Triangle of rationals r(m,n):= A120072(m,n)/A120073(m,n), m>=2, 1<= n <= m-1. r(m,n) = A120070(m,n)/(m^2*n^2) = 1/n^2 - 1/m^2. Used for spectrum of H-atom (see below). m\n 1 2 3 4 5 6 7 8 9 10 ... 2 3/4 0 0 0 0 0 0 0 0 0 3 8/9 5/36 0 0 0 0 0 0 0 0 4 15/16 3/16 7/144 0 0 0 0 0 0 0 5 24/25 21/100 16/225 9/400 0 0 0 0 0 0 6 35/36 2/9 1/12 5/144 11/900 0 0 0 0 0 7 48/49 45/196 40/441 33/784 24/1225 13/1764 0 0 0 0 8 63/64 15/64 55/576 3/64 39/1600 7/576 15/3136 0 0 0 9 80/81 77/324 8/81 65/1296 56/2025 5/324 32/3969 17/5184 0 0 10 99/100 6/25 91/900 21/400 3/100 4/225 51/4900 9/1600 19/8100 0 11 120/121 117/484 112/1089 105/1936 96/3025 85/4356 72/5929 57/7744 40/9801 21/12100 . . . ###################################################################################################################################### Row sums give the rationals r(m):=A120076(m)/A120077(m), m>=2: [3/4, 37/36, 169/144, 4549/3600, 4769/3600, 241481/176400, 989549/705600, 9072541/6350400, 1841321/1270080, 225467009/153679680,...]. r(m)= Zeta(2;m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2;n):=sum(1/k^2,k=1..n). See the W.Lang link under A103345. O.g.f. for r(m), m>=2: R(x):= ln(1-x) + polylog(2,x)/(1-x) = ln(1-x) + (dilog(1-x))/(1-x). ####################################################################################################################################### The generating function for the column n numbers, R(n,x):= sum(r(m,n)*x^m,m=n+1..infinity), n>=1, has the following form: R(n,x) = -dilog(1-x) + x*P(n,x))/(A(n)*(n^2)*(1-x)), n>=1, where dilog(1-x)=polylog(2,x), the sequence A(n)=[1,1,4,9,144,100,3600,11025,78400,63504,...] = A027451(n) (conjecture), n>=1. See below for more on A(n). and the first ten polynomials P(n,x) are (see the coefficient triangle A120078): n=1: P(1,x) = 1, n=2: P(2,x) = 4-3*x, n=3: P(3,x) = 36-27*x-5*x^2, n=4: P(4,x) = 144-108*x-20*x^2-7*x^3, n=5: P(5,x) = 3600-2700*x-500*x^2-175*x^3-81*x^4, n=6: P(6,x) = 3600-2700*x-500*x^2-175*x^3-81*x^4-44*x^5, n=7: P(7,x) = 176400-132300*x-24500*x^2-8575*x^3-3969*x^4-2156*x^5-1300*x^6, n=8: P(8,x) = 705600-529200*x-98000*x^2-34300*x^3-15876*x^4-8624*x^5-5200*x^6-3375*x^7, n=9: P(9,x) = 6350400-4762800*x-882000*x^2-308700*x^3-142884*x^4-77616*x^5-46800*x^6-30375*x^7-20825*x^8, n=10: P(10,x) = 6350400-4762800*x-882000*x^2-308700*x^3-142884*x^4-77616*x^5-46800*x^6-30375*x^7-20825*x^8-14896*x^9 ... These integer polynomials P(n,x) of degree n-1 can be expessed in terms of partial sums of the k=2 polylog defined by polylog(2;x,n):=sum((x^m)/m^2,m=1..n), n>=1, in the follwing way. P(n,x)/A(n) = x^n + n^2*(1-x)*polylogknx(2,n,x)/x, with the numbers A(n) from above. If one uses x^n + n^2*(1-x)*polylogknx(2,n,x)/x = (n^2)*(1 - sum(((2*k+1)/(k*(k+1)))^2)*x^k,k=1..n-1), n>=1, with the sum put to zero for n=1, it becomes clear that P(n,x) becomes an integer polynomial if one takes A(n)=LCM((1*2)^2, (2*3)^2,...,(n-2)*(n-1)^2, (n-1)^2) = LCM(seq((k*(k+1)))^2,k=1..n-1)) = (LCM(seq(k*(k+1)),k=1..n-1))^2, n>=2. A(1):=1. Therefore A(n)=[1,1,4,36,144,3600,3600,176400,705600,6350400,...],n>=1, the squares of [1,1,2,6,12,60,60,420,840,2520,...] = A003418(n-1). This follows from the fact that (2*k+1) (odd) never divides (k*(k+1))^2 (even), and only the last term in the sum is divisible by n^2. Therefore, A(n) is the least common multiple (LCM) of the denominators of the entries in row number n, n>=1, of the rational triangle whose row polynomials are p(n,x):= x^n + n^2*(1-x)*polylogknx(2,n,x)/x = (n^2)*(1 - sum(((2*k+1)/(k*(k+1)))^2)*x^k,k=1..n-1), n>=1, with the sum put to zero for n=1. ################################################################################################################################# The triangle for the numerators is A120072(m,n): a(m,n) tabl head (triangle) for A120072 m\n 1 2 3 4 5 6 7 8 9 10 ... 2 3 0 0 0 0 0 0 0 0 0 3 8 5 0 0 0 0 0 0 0 0 4 15 3 7 0 0 0 0 0 0 0 5 24 21 16 9 0 0 0 0 0 0 6 35 2 1 5 11 0 0 0 0 0 7 48 45 40 33 24 13 0 0 0 0 8 63 15 55 3 39 7 15 0 0 0 9 80 77 8 65 56 5 32 17 0 0 10 99 6 91 21 3 4 51 9 19 0 11 120 117 112 105 96 85 72 57 40 21 . . . The row sums give [3, 13, 25, 70, 54, 203, 197, 340, 303, 825, ...] = A120074(m), m>=2. ########################################################################################## The triangle for the denominators is A120073(m,n): a(m,n) tabl head (triangle) for A120073 m\n 1 2 3 4 5 6 7 8 9 10 ... 2 4 0 0 0 0 0 0 0 0 0 3 9 36 0 0 0 0 0 0 0 0 4 16 16 144 0 0 0 0 0 0 0 5 25 100 225 400 0 0 0 0 0 0 6 36 9 12 144 900 0 0 0 0 0 7 49 196 441 784 1225 1764 0 0 0 0 8 64 64 576 64 1600 576 3136 0 0 0 9 81 324 81 1296 2025 324 3969 5184 0 0 10 100 25 900 400 100 225 4900 1600 8100 0 11 121 484 1089 1936 3025 4356 5929 7744 9801 12100 . . . The row sums give: [4, 45, 176, 750, 1101, 4459, 6080, 13284, 16350, 46585,....] = A120075(m), m>=2. ############################################################################################################## ############################################################################################################## The frequencies of the spectral lines of the H-atom are (approximately): nu(m,n) = r(m,n) * 3.287*10^{15} s^{-1} = r(m,n) * 3.287 PHz, P=Peta for 10^15, Hz=Hertz=1/s. The corresponding energies: E(m,n) = r(m,n)*13.599 eV, (eV= electron volts, 1 eV = 1.602 176 53(14)*10^{-19} J , J=Joule). The corresponding wave lengths: lambda(m,n) = (1/r(m,n)) * 91.196 nm, (1 nm = 10^{-9} m) ##################################################################################################################### The n=1,2,3,4 and 5 series, for m>=2,3,4,5 and 6, bear the names of Lyman, Balmer, Paschen, Brackett and Pfund, respectively. e.g.: n=2 (Balmer series) partly in the visible range (approximate values): lambda(3,2) = 657 nm (red), lambda(4,2) = 486 nm (green), lambda(5,2) = 434 nm (blue), lambda(6,2) = 410 nm (violet), lambda(7,2) = 397 nm (violet), lambda(8,2) = 389 nm (violet), lambda(9,2) = 384 nm (ultraviolet), ... The corresponding (approximate) frequencies are (T=Tera = 10^{12}): nu(3,2) = 457 THz, nu(4,2) = 617*THz, nu(5,2) = 731*THz, nu(6,2) =755*THz, nu(7,2) = 771*THz, nu(8,2) = 782*THz, nu(9,2) = 790*THz, ... The corresponding enrgies are (4 digits): 1.89*eV, 2.55*eV, 2.86*eV, 3.02*eV, 3.12*eV, 3.19*eV, 3.23*eV, ... ############################################ Lyman series (n=1, m>=2) in the ultraviolet: frequencies (3 digits): 2.47*PHz, 2.92*PHz, 3.08*PHz, 3.16*PHz, 3.20*PHz, 3.22*PHz, 3.24*PHz, 3.25*PHz, 3.26*PHz, 3.26*PHz, 3.27*PHz, ... wavelengths (4 digits): 121.6*nm, 102.6*nm, 97.28*nm, 95.00*nm, 93.81*nm, 93.10*nm, 92.65*nm, 92.34*nm, 92.12*nm, 91.96*nm, 91.84*nm,... energies (4 digits): 10.20*eV, 12.09*eV, 12.75*eV, 13.06*eV, 13.22*eV, 13.32*eV, 13.39*eV, 13.43*eV, 13.46*eV, 13.49*eV, 13.51*eV, ... ########################################### Paschen series (n=3, m>=4) in the infrared: frequencies (3 digits): 160 THz, 234 THz, 274 THz, 298 THz, 314 THz, 325 THz, 333 THz, 338 THz, 343 THz, 346 THz, ... wavelengths (4 digits): 1876.*nm, 1282.*nm, 1094.*nm, 1005.*nm, 955.1*nm, 923.4*nm, 902.0*nm, 886.8*nm, 875.5*nm, 867.0*nm,... energies (3 digits): .661*eV, .967*eV, 1.13*eV, 1.23*eV, 1.30*eV, 1.34*eV, 1.37*eV, 1.40*eV, 1.42*eV, 1.43*eV, ... ########################################## Brackett series (n=4, m>=5) in the infrared: frequencies (approximate): 74*THz, 114*THz, 138*THz, 154*THz, 165*THz, 173*THz, 178*THz, 183*THz, 186*THz, ... wavelengths (4 digits, 10^3 nm = 1 micro (mu) m) : 4053*nm, 2627*nm, 2167*nm, 1946*nm, 1818*nm, 1737*nm, 1682*nm, 1642*nm, 1612*nm, ... energies (3 digits) meV (milli eV): 306*meV, 472*meV, 572*meV, 638*meV, 682*meV, 714*meV, 738*meV, 756*meV, 769*meV, ... ########################################## Pfund series (n=5, m>=6) in the infrared: frequencies (approximate): 40.2*THz, 64.5*THz, 80.2*THz, 91.0*THz, 98.7*THz, 104*THz, 109*THz, ... wavelengths (4 digits, 10^3 nm = 1 micro (mu) m) : 7462*nm, 4655*nm, 3742*nm, 3298*nm, 3040*nm, 2874*nm, 2759*nm, 2676*nm, ... energies (3 digits) meV (milli eV): 166*meV, 266*meV, 332*meV, 376*meV, 408*meV, 432*meV, 450*meV, 463*meV, ... ############################################### e.o.f.##############################################################