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%I #72 Mar 14 2024 16:29:44
%S 1,1,3,1,15,9,1,63,108,27,1,255,945,594,81,1,1023,7380,8775,2835,243,
%T 1,4095,54729,109890,63180,12393,729,1,16383,395388,1263087,1151010,
%U 387828,51030,2187,1,65535,2816865,13817034,18752391,9658278,2133054,201204,6561,1,262143,19914660,146620935,285232185,210789621,69502860,10825650,767637,19683
%N Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
%C For Sheffer triangles (infinite lower triangular exponential convolution matrices) see the W. Lang link under A006232, with references).
%C The e.g.f. for the sequence of column m is (Sheffer property) exp(x)*(exp(3*x) - 1)^m/m!.
%C This is a generalization of the Sheffer triangle Stirling2(n, m) = A048993(n, m) denoted by (exp(x), exp(x)-1), which could be named S2[1,0].
%C The a-sequence for this Sheffer triangle has e.g.f. 3*x/log(1+x) and is 3*A006232(n)/ A006233(n) (Cauchy numbers of the first kind).
%C The z-sequence has e.g.f. (3/(log(1+x)))*(1 - 1/(1+x)^(1/3)) and is A284857(n) / A284858(n).
%C The main diagonal gives A000244.
%C The row sums give A284859. The alternating row sums give A284860.
%C The triangle appears in the o.g.f. G(n, x) of the sequence {(1 + 3*m)^n}_{m>=0}, as G(n, x) = Sum_{m=0..n} T(n, m)*m!*x^m/(1-x)^(m+1), n >= 0. Hence the corresponding e.g.f. is, by the linear inverse Laplace transform, E(n, t) = Sum_{m >=0}(1 + 3*m)^n t^m/m! = exp(t)*Sum_{m=0..n} T(n, m)*t^m.
%C The corresponding Euler triangle with reversed rows is rEu(n, k) = Sum_{m=0..k} (-1)^(k-m)*binomial(n-m, k-m)*T(n, k)*k!, 0 <= k <= n. This is A225117 with row reversion.
%C The first column k sequences divided by 3^k are A000012, A002450 (with a leading 0), A016223, A021874. For the e.g.f.s and o.g.f.s see below. - _Wolfdieter Lang_, Apr 09 2017
%C From _Wolfdieter Lang_, Aug 09 2017: (Start)
%C The general row polynomials R(d,a;n,x) = Sum_{k=0..n} T(d,a;n,m)*x^m of the Sheffer triangle S2[d,a] satisfy, as special polynomials of the Boas-Buck class, the identity (see the reference, and we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function)
%C (E_x - n*1)*R(d,a;n,x) = - n*a*R(d,a;n-1,x) - Sum_{k=0..n-1} binomial(n, k+1)*(-d)^(k+1)*Bernoulli(k+1)*E_x*R(d,a;n-1-k,x), with E_x = x*d/dx (Euler operator).
%C This entails a recurrence for the sequence of column m, for n > m:
%C T(d,a;n,m) = (1/(n - m))*[(n/2)*(2*a + d*m)*T(d,a;n-1,m) + m*Sum_{p=m..n-2} binomial(n,p)(-d)^(n-p)*Bernoulli(n-p)*T(d,a;p,m)], with input T(d,a;n,n) = d^n. For the present [d,a] = [3,1] case see the formula and example sections below. - _Wolfdieter Lang_, Aug 09 2017 (End)
%C The inverse of this triangular Sheffer matrix S2[3,1] is S1[3,1] with rational elements S1[3,1](n, k) = (-1)^(n-k)*A286718(n, k)/3^k. - _Wolfdieter Lang_, Nov 15 2018
%C Named after the American mathematician Isador Mitchell Sheffer (1901-1992). - _Amiram Eldar_, Jun 19 2021
%D Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
%D Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
%H Michael De Vlieger, <a href="/A282629/b282629.txt">Table of n, a(n) for n = 0..11475</a>, rows n = 0..150, flattened.
%H Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 9.
%H Wolfdieter Lang, <a href="http://arXiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers</a>, arXiv:math/1707.04451 [math.NT], July 2017.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%F A nontrivial recurrence for the column m=0 entries T(n, 0) = 1 from the z-sequence given above: T(n,0) = n*Sum_{j=0..n-1} z(j)*T(n-1,j), n >= 1, T(0, 0) = 1.
%F Recurrence for column m >= 1 entries from the a-sequence given above: T(n, m) = (n/m)*Sum_{j=0..n-m} binomial(m-1+j, m-1)*a(j)*T(n-1, m-1+j), m >= 1.
%F Recurrence for row polynomials R(n, x) (Meixner type): R(n, x) = ((3*x+1) + 3*x*d_x)*R(n-1, x), with differentiation d_x, for n >= 1, with input R(0, x) = 1.
%F T(n, m) = Sum_{k=0..m} binomial(m,k)*(-1)^(k-m)*(1 + 3*k)^n/m!, 0 <= m <= n.
%F E.g.f. of triangle: exp(z)*exp(x*(exp(3*z)-1)) (Sheffer type).
%F E.g.f. for sequence of column m is exp(x)*((exp(3*x) - 1)^m)/m! (Sheffer property).
%F From _Wolfdieter Lang_, Apr 09 2017: (Start)
%F Standard three-term recurrence: T(n, m) = 0 if n < m, T(n,-1) = 0, T(0, 0) = 1, T(n, m) = 3*T(n-1, m-1) + (1+3*m)*T(n-1, m) for n >= 1. From the T(n, m) formula. Compare with the recurrence of S2[3,2] given in A225466.
%F The o.g.f. for sequence of column m is 3^m*x^m/Product_{j=0..m} (1 - (1+3*j)*x). (End)
%F In terms of Stirling2 = A048993: T(n, m) = Sum_{k=0..n} binomial(n, k)* 3^k*Stirling2(k, m), 0 <= m <= n. - _Wolfdieter Lang_, Apr 13 2017
%F Boas-Buck recurrence for column sequence m: T(n, m) = (1/(n - m))*[(n/2)*(2 + 3*m)*T(n-1, m) + m*Sum_{p=m..n-2} binomial(n,p)(-3)^(n-p)*Bernoulli(n-p)*T(p, m)], for n > m >= 0, with input T(m, m) = 3^m. See a comment above. - _Wolfdieter Lang_, Aug 09 2017
%e The triangle T(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9
%e 0: 1
%e 1: 1 3
%e 2: 1 15 9
%e 3: 1 63 108 27
%e 4: 1 255 945 594 81
%e 5: 1 1023 7380 8775 2835 243
%e 6: 1 4095 54729 109890 63180 12393 729
%e 7: 1 16383 395388 1263087 1151010 387828 51030 2187
%e 8: 1 65535 2816865 13817034 18752391 9658278 2133054 201204 6561
%e 9: 1 262143 19914660 146620935 285232185 210789621 69502860 10825650 767637 19683
%e ...
%e ------------------------------------------------------------------------------------
%e Nontrivial recurrence for m=0 column from z-sequence:T(4,0) = 4*(1*1 + 63*(-1/6) + 108*(11/54) + 27*(-49/108)) = 1.
%e Recurrence for m=2 column from a-sequence: T(4, 2) = (4/2)*(1*63*3 + 2*108*(3/2) + 3*27*(-3/6)) = 945.
%e Recurrence for row polynomial R(3, x) (Meixner type): ((3*x + 1) + 3*x*d_x)*(1 + 15*x + 9*x^2) = 1 + 63*x + 108*x^2 + 27*x^3.
%e E.g.f. and o.g.f. of n = 1 powers {(1 + 3*m)^1}_{m>=0} A016777: E(1, x) = exp(x) * (T(1, 0) + T(1, 1)*x) = exp(x)*(1+3*x). O.g.f.: G(1, x) = T(1, 0)*0!/(1-x) + T(1, 1)*1!*x/(1-x)^2 = (1+2*x)/(1-x)^2.
%e Boas-Buck recurrence for column m = 2, and n = 4: T(4, 2) =(1/2)*[2*(2 + 3*2)*T(3, 2) + 2*6*(-3)^2*bernoulli(2)*T(2, 2))] = (1/2)*(16*108 + 12*9*(1/6)*9) = 945. - _Wolfdieter Lang_, Aug 09 2017
%t Table[Sum[Binomial[m, k] (-1)^(k - m) (1 + 3 k)^n/m!, {k, 0, m}], {n, 0, 9}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 08 2017 *)
%o (PARI) T(n, m) = sum(k=0, m, binomial(m, k) * (-1)^(k - m) * (1 + 3*k)^n/m!);
%o for(n=0, 9, for(m=0, n, print1(T(n, m),", ");); print();) \\ _Indranil Ghosh_, Apr 08 2017
%Y Cf. A000012, A000244, A006232/A006233, A016777, A024036, A111577, A225117, A225466, A284857, A284858, A284859, A284860, A284861, A286718.
%K nonn,easy,tabl
%O 0,3
%A _Wolfdieter Lang_, Apr 03 2017
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