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%I #35 Jan 27 2020 18:36:39
%S 1,2,1,12,12,1,120,180,30,1,1680,3360,840,56,1,30240,75600,25200,2520,
%T 90,1,665280,1995840,831600,110880,5940,132,1,17297280,60540480,
%U 30270240,5045040,360360,12012,182,1,518918400,2075673600,1210809600,242161920,21621600,960960,21840,240,1,17643225600,79394515200,52929676800,12350257920,1323241920,73513440,2227680,36720,306,1
%N Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].
%C s(n,x) := Sum_{m=0..n} T(n,m)*x^m are monic polynomials satisfying s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} A048786(n,m)*x^m (row polynomials of triangle A048786) and p(0,x)=1.
%C In the umbral calculus (see the Roman reference, p. 21) the s(n,x) are called Sheffer polynomials for(1/sqrt(1+4*t),t/(1+4*t)). Here the Sheffer notation differs. See the W. Lang link under A006232.
%C For the general L[d,a] triangles see A286724, also for references.
%C This is the generalized signless Lah number triangle L[4,1], the Sheffer triangle ((1 - 4*t)^(-1/2), t/(1 - 4*t)). It is defined as transition matrix
%C risefac[4,1](x, n) = Sum_{m=0..n} L[4,1](n, m)*fallfac[4,1](x, m), where risefac[4,1](x, n) := Product_{0..n-1} (x + (1 + 4*j)) for n >= 1 and risefac[4,1](x, 0) := 1, and fallfac[4,1](x, n) := Product_{0..n-1} (x - (1 + 4*j)) for n >= 1 and fallfac[4,1](x, 0) := 1.
%C In matrix notation: L[4,1] = S1phat[4,1]*S2hat[4,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A290319 and A111578 (but here with offsets 0), respectively.
%C The a- and z-sequences for this Sheffer matrix have e.g.f.s Ea(t) = 1 + 4*t and Ez(t) = (1 + 4*t)*(1 - (1 + 4*t)^(-1/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292220(n). See the W. Lang link on a- and z-sequences there.
%C The inverse matrix T^(-1) = L^(-1)[4,1] is Sheffer ((1 + 4*t)^(-1/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
%C fallfac[4,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,1](x, m), n >= 0.
%C Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A091042(d, m)*x^m/(1 - x)^{2*d + 1}, for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(n + d),2*d)}_{n >= 0}. See the second W. Lang link on how to compute o.g.f.s of diagonal sequences of general Sheffer triangles. - _Wolfdieter Lang_, Oct 12 2017
%D S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
%H Michael De Vlieger, <a href="/A048854/b048854.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H Orli Herscovici, Ross G. Pinsky, <a href="https://pdfs.semanticscholar.org/8fe9/91ddcd917fdcce4a5d5de8fbf4a61859c25a.pdf">An identity involving stirling numbers of both kinds and its connection to right-to-left minima of certain set partitions</a>, (2019).
%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, section C) 4.
%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1708.01421">On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
%F T(n, m) = (n!/m!)*A046521(n, m) = (n!/m!)* binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0, a(n, m) := 0, n < m.
%F Sum_{n>=0, k>=0} T(n, k)*x^n*y^k/(2*n)! = exp(x)*cosh(sqrt(x*y)). - _Vladeta Jovovic_, Feb 21 2003
%F T(n, m) = L[4,1](n,m) = Sum_{k=m..n} A290319(n, k)*A111578(k+1, m+1), 0 <= m <= n.
%F E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
%F (1 - 4*t)^(-1/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
%F E.g.f. of column m: (1 - 4*t)^(-1/2)*(t/(1 - 4*t))^m/m!, m >= 0.
%F Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 4*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), n >= 1, T(0, 0) = 0, from the a-sequence {1, 4 repeat(0)} and the z(j) = 2*A292220(j) (see above).
%F Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 3)*T(n-1, m) - 8*(n-1)*(2*n - 3)*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 + 4*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} (-4)^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) = [(2 + x)*1 + 8*(1 + x)*D_x + 16*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))*(2 + 4*m)*Sum_{p=0..n-1-m} 4^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
%F Explicit form (from the o.g.f.s of diagonal sequences): ((2*(n-m))!/(n-m)!)*binomial(2*n,2*(n-m)), n >= m >= 0, and vanishing for n < m. - _Wolfdieter Lang_, Oct 12 2017
%e The triangle T(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 ...
%e 0: 1
%e 1: 2 1
%e 2: 12 12 1
%e 3: 120 180 30 1
%e 4: 1680 3360 840 56 1
%e 5: 30240 75600 25200 2520 90 1
%e 6: 665280 1995840 831600 110880 5940 132 1
%e 7: 17297280 60540480 30270240 5045040 360360 12012 182 1
%e 8: 518918400 2075673600 1210809600 242161920 21621600 960960 21840 240 1
%e ...
%e n = 9: 17643225600 79394515200 52929676800 12350257920 1323241920 73513440 2227680 36720 306 1,
%e n = 10: 670442572800 3352212864000 2514159648000 670442572800 83805321600 5587021440 211629600 4651200 58140 380 1.
%e ...
%e Recurrence from a-sequence: T(4, 2) = 2*T(3, 1) + 4*4*T(3, 2) = 2*180 + 16*30 = 840.
%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*(2*120 + 2*180 - 8*30 + 60*1) = 1680.
%e Four term recurrence: T(4, 2) = T(3, 1) + 2*13*T(3, 2) - 8*3*5*T(2, 2) = 180 + 26*30 - 120*1 = 840.
%e Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(12 + 12*x + 1*x^2) = (12 + 2*x) - 4*2 = 2*(2 + x).
%e Sheffer recurrence for R(3, x): [(2 + x) + 8*(1 + x)*D_x + 16*x*(D_x)^2] (12 + 12*x + 1*x^2) = (2 + x)*(12 + 12*x + x^2) + 8*(1 + x)*(12 + 2*x) + 16*2*x = 120 + 180*x + 30*x^2 + x^3 = R(3, x).
%e Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*10/2)*(1*30/3! + 4*1/2!) = 840.
%e Diagonal sequence d = 2: {12, 180, 840 ...} has o.g.f. 12*(1 + 10*x + 5*x^2)/(1 - x)^5 (see A001813(2) and row n=2 of A091042) generating
%e {12*binomial(2*(n + 2), 4)}_{n >= 0}. - _Wolfdieter Lang_, Oct 12 2017
%p A290604_row := proc(n) exp(x*t/(1-4*t))/sqrt(1-4*t): series(%, t, n+2): seq(n!*coeff(coeff(%,t,n),x,j), j=0..n) end: seq(A290604_row(n), n=0..9); # _Peter Luschny_, Sep 23 2017
%t T[n_, m_] := n!/m! * Binomial[2*n, n] * Binomial[n, m] / Binomial[2*m, m]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 05 2013 *)
%t T[0, 0] = 1; T[-1, _] = T[_, -1] = 0; T[n_, m_] /; n < m = 0; T[n_, m_] := T[n, m] = T[n-1, m-1] + 2*(4*n-3)*T[n-1, m] - 8*(n-1)*(2*n-3)*T[n-2, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 23 2017 *)
%Y Related to triangle A046521. Cf. A048786. a(n, 0) = A001813.
%Y A111578, A271703 L[1,0], A286724 L[2,1], A290319, A290596 L[3,1], A290597 L[3,2], A292220.
%Y The diagonal sequences are: A000012, 2*A000384(n+1), 12*A053134, 120*A053135, 1680*A053137, ... - _Wolfdieter Lang_, Oct 12 2017
%K easy,nonn,tabl
%O 0,2
%A _Wolfdieter Lang_
%E Name changed, after merging my newer duplicate, from _Wolfdieter Lang_, Oct 10 2017
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