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%I #9 Sep 16 2018 21:40:38
%S 1,4,1,28,14,1,280,210,30,1,3640,3640,780,52,1,58240,72800,20800,2080,
%T 80,1,1106560,1659840,592800,79040,4560,114,1,24344320,42602560,
%U 18258240,3043040,234080,8778,154,1,608608000,1217216000,608608000,121721600,11704000,585200,15400,200,1,17041024000,38342304000,21909888000,5112307200,589881600,36867600,1293600,25200,252,1,528271744000,1320679360000,849008160000,226402176000,30477216000,2285791200,100254000,2604000,39060,310,1
%N Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,2].
%C For the general L[d,a] triangles see A286724, also for references.
%C This is the generalized signless Lah number triangle L[3,2], the Sheffer triangle ((1 - 3*t)^(-4/3), t/(1 - 3*t)). It is defined as transition matrix risefac[3,2](x, n) = Sum_{m=0..n} L[3,2](n, m)*fallfac[3,2](x, m), where risefac[3,2](x, n):= Product_{0..n-1} (x + (2 + 3*j)) for n >= 1 and risefac[3,2](x, 0) := 1, and fallfac[3,2](x, n):= Product_{0..n-1} (x - (2 + 3*j)) for n >= 1 and fallfac[3,2](x, 0) := 1.
%C In matrix notation: L[3,2] = S1phat[3,2]*S2hat[3,2] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A225470 and A225468, respectively.
%C The a- and z-sequences for this Sheffer matrix have e.g.f.s 1 + 3*t and (1 + 3*t)*(1 - (1 + 3*t)^(-4/3))/t, respectively. That is, a = {1, 3, repeat(0)} and z(n) = A290603(n)/A038500(n+1). See a W. Lang link under A006232 for these types of sequences with a reference, and also the present link, eq. (142).
%C The inverse matrix T^(-1) = L^(-1)[3,2] is Sheffer ((1 + 3*t)^(-4/3), t/(1 + 3*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
%C fallfac[3,2](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,2](x, m), n >= 0.
%D Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.
%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.
%F T(n, m) = L[3,2](n,m) = Sum_{k=m..n} A225470(n, k) * A225468(k, m), 0 <= m <= n.
%F E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
%F (1 - 3*t)^(-4/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
%F E.g.f. of column m: (1 - 3*t)^(-4/3)*(t/(1 - 3*t))^m/m!, m >= 0.
%F Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 3*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0}^(n-1) z(j)*T(n-1, j), from the a-sequence {1, 3 repeat(0)} and the z-sequence given above.
%F Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(3*n - 1)*T(n-1, m) - 3*(n-1)*(3*n - 2)*T(n-2, m), n >= m >= 0, with T(0, 0) =1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
%F Meixner type identity for (monic) row polynomials: (D_x/(1 + 3*D_x)) * R(n, x) = n*R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-3)^k*{D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
%F General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
%F R(n, x) = [(4 + x)*1 + 6*(2 + x)*D_x + 3^2*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
%F Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(4 + 3*m)*Sum_{p=0..n-1-m} 3^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
%e The triangle T(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 ...
%e 0: 1
%e 1: 4 1
%e 2: 28 14 1
%e 3: 280 210 30 1
%e 4: 3640 3640 780 52 1
%e 5: 58240 72800 20800 2080 80 1
%e 6: 1106560 1659840 592800 79040 4560 114 1
%e 7: 24344320 42602560 18258240 3043040 234080 8778 154 1
%e 8: 608608000 1217216000 608608000 121721600 11704000 585200 15400 200 1
%e ...
%e n = 9: 17041024000 38342304000 21909888000 5112307200 589881600 36867600 1293600 25200 252 1,
%e n = 10: 528271744000 1320679360000 849008160000 226402176000 30477216000 2285791200 100254000 2604000 39060 310 1.
%e ...
%e Recurrence from a-sequence: T(4, 2) = (4/2)*T(3, 1) + 3*4*T(3, 2) = 2*210 + 12*30 = 780.
%e Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2) + z(3)*T(3, 3)) = 4*(4* 280 - 2*210 + (28/3)*30 - 70*1) = 3640.
%e Four term recurrence: T(4, 2) = T(3, 1) + 2*11*T(3, 2) - 3*3*10*T(2, 2) = 210 + 22*30 - 90*1 = 780.
%e Meixner type identity for n = 2: (D_x - 3*(D_x)^2)*(28 + 14*x + x^2) = (14 + 2*x) - 3*2 = 2*(4 + x).
%e Sheffer recurrence for R(3, x): [(4 + x) + 6*(2 + x)*D_x + 9*x*(D_x)^2] (28 + 14*x + x^2) = (4 + x)*(28 + 14*x + x^2) + 6*(2 + x)*(14 + 2*x) + 9*2*x= 280 + 210*x + 30*x^2 + x^3 = R(3, x).
%e Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*(4 + 3*2)/2)*(1*30/3! + 3*1/2!) = 780.
%Y Cf. A007559(n+1) (column m=0), A225468, A225470, A271703 L[1,0], A286724 L[2,1], A290596, L[3,1], A290603.
%K nonn,tabl,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 13 2017
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