|
%I M4855 N2075 #125 Oct 18 2022 19:10:58
%S 1,12,90,560,3150,16632,84084,411840,1969110,9237800,42678636,
%T 194699232,878850700,3931426800,17450721000,76938289920,337206098790,
%U 1470171918600,6379820115900,27569305764000,118685861314020,509191949220240,2177742427450200,9287309860732800
%N a(n) = binomial(2*n+1,n)*(n+1)^2.
%C Coefficients for numerical differentiation.
%C Take the first n integers 1,2,3..n and find all combinations with repetitions allowed for the first n of them. Find the sum of each of these combinations to get this sequence. Example for 1 and 2: 1,2,1+1,1+2,2+2 gives sum of 12=a(2). - _J. M. Bergot_, Mar 08 2016
%C Let cos(x) = 1 -x^2/2 +x^4/4!-x^6/6!.. = Sum_i (-1)^i x^(2i)/(2i)! be the standard power series of the cosine, and y = 2*(1-cos(x)) = 4*sin^2(x/2) = x^2 -x^4/12 +x^6/360 ...= Sum_i 2*(-1)^(i+1) x^(2i)/(2i)! be a closely related series. Then this sequence represents the reversion x^2 = Sum_i 1/a(i) *y^(i+1). - _R. J. Mathar_, May 03 2022
%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002544/b002544.txt">Table of n, a(n) for n=0..200</a>
%H W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H A. Petojevic and N. Dapic, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf">The vAm(a,b,c;z) function</a>, Preprint 2013.
%H H. E. Salzer, <a href="http://dx.doi.org/10.1002/sapm1943221115">Coefficients for numerical differentiation with central differences</a>, J. Math. Phys., 22 (1943), 115-135.
%H H. E. Salzer, <a href="/A002457/a002457_2.pdf">Coefficients for numerical differentiation with central differences</a>, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy]
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%H R. Shenton and A. W. Kemp, <a href="https://doi.org/10.1016/0377-0427(89)90308-7">An S-fraction and ln^2(1+x)</a>, Journal of Computational and Applied Mathematics, 26 (1989) 367-370 North-Holland.
%H T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/21/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 21.
%H Mats Vermeeren, <a href="http://arxiv.org/abs/1506.05288">A dynamical solution to the Basel problem</a>, arXiv preprint arXiv:1506.05288 [math.CA], 2015.
%F G.f.: (1 + 2x)/(1 - 4x)^(5/2).
%F a(n-1) = sum(i_1 + i_2 + ... + i_n) where the sum is over 0 <= i_1 <= i_2 <= ... <= i_n <= n; a(n) = (n+1)^2 C(2n+1, n). - _David Callan_, Nov 20 2003
%F a(n) = (n+1)^2 * binomial(2*n+2,n+1)/2. - _Zerinvary Lajos_, May 31 2006
%F Asymptotics: a(n)-> (1/64) * (128*n^2+176*n+41) * 4^n * n^(-1/2)/(sqrt(Pi)), for n->infinity. - _Karol A. Penson_, Aug 05 2013
%F G.f.: 2F1(3/2,2;1;4x). - _R. J. Mathar_, Aug 09 2015
%F a(n) = A002457(n)*(n+1). - _R. J. Mathar_, Aug 09 2015
%F a(n) = A000217(n)*A000984(n). - _J. M. Bergot_, Mar 10 2016
%F a(n-1) = A001791(n)*n*(n+1)/2. - _Anton Zakharov_, Jul 04 2016
%F From _Ilya Gutkovskiy_, Jul 04 2016: (Start)
%F E.g.f.: ((1 + 2*x)*(1 + 8*x)*BesselI(0,2*x) + 2*x*(3 + 8*x)*BesselI(1,2*x))*exp(2*x).
%F Sum_{n>=0} 1/a(n) = Pi^2/9 = A100044. (End)
%F From _Peter Bala_, Apr 18 2017: (Start)
%F With x = y^2/(1 + y) we have log^2(1 + y) = Sum_{n >= 0} (-1)^n*x^(n+1)/a(n). See Shenton and Kemp.
%F Series reversion ( Sum_{n >= 0} (-1)^n*x^(n+1)/a(n) ) = Sum_{n >= 1} 2*x^n/(2*n)! = Sum_{n >= 1} x^n/A002674(n). (End)
%F D-finite with recurrence n^2*a(n) -2*(n+1)*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021
%F Sum_{n>=0} (-1)^n/a(n) = 4*arcsinh(1/2)^2 = A202543^2. - _Amiram Eldar_, May 14 2022
%p seq((n+1)^2*(binomial(2*n+2, n+1))/2, n=0..29); # _Zerinvary Lajos_, May 31 2006
%t Table[Binomial[2n+1,n](n+1)^2,{n,0,20}] (* _Harvey P. Dale_, Mar 23 2011 *)
%o (PARI) a(n)=binomial(2*n+1,n)*(n+1)^2
%o (PARI) x='x+O('x^99); Vec((1+2*x)/(1-4*x)^(5/2)) \\ _Altug Alkan_, Jul 09 2016
%o (Python)
%o from sympy import binomial
%o def a(n): return binomial(2*n + 1, n)*(n + 1)**2 # _Indranil Ghosh_, Apr 18 2017
%Y Cf. A085373, A002457, A002674, A202543.
%Y Equals A002736/2.
%Y A diagonal of A331430.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
| 