This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A290596 #13 Sep 23 2017 04:40:26
%S A290596 1,2,1,10,10,1,80,120,24,1,880,1760,528,44,1,12320,30800,12320,1540,
%T A290596 70,1,209440,628320,314160,52360,3570,102,1,4188800,14660800,8796480,
%U A290596 1832600,166600,7140,140,1,96342400,385369600,269758720,67439680,7663600,437920,12880,184,1,2504902400,11272060800,9017648640,2630147520,358656480,25618320,1004640,21528,234,1,72642169600,363210848000,326889763200,108963254400,17335063200,1485862560,72836400,2081040,33930,290,1
%N A290596 Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].
%C A290596 For the general L[d,a] triangles see A286724, also for references.
%C A290596 This is the generalized signless Lah number triangle L[3,1], the Sheffer triangle ((1 - 3*t)^(-2/3), t/(1 - 3*t)). It is defined as transition matrix
%C A290596 risefac[3,1](x, n) = Sum_{m=0..n} L[3,1](n, m)*fallfac[3,1](x, m), where risefac[3,1](x, n):= Product_{0..n-1} (x + (1 + 3*j)) for n >= 1 and risefac[3,1](x, 0) := 1, and fallfac[3,1](x, n):= Product_{0..n-1} (x - (1 + 3*j)) for n >= 1 and fallfac[3,1](x, 0) := 1.
%C A290596 In matrix notation: L[3,1] = S1phat[3,1]*S2hat[3,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A286718 and A111577 (but here with offsets 0), respectively.
%C A290596 The a- and z-sequences for this Sheffer matrix has e.g.f.s Ea(t) = 1 + 3*t and (Ez(t) = (1 + 3*t)*(1 - (1 + 3*t)^(-2/3))/t, respectively. That is, a = {1, 3, repeat(0)} and z(n) = A290597(n)/A038500(n+1). For the proof see the second W. Lang link. See also a W. Lang link under A006232 for Sheffer a- and z-sequences with references (in the Riordan case).
%C A290596 The inverse matrix T^(-1) = L^(-1)[3,1] is Sheffer ((1 + 3*t)^(-2/3), t/(1 + 3*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
%C A290596 fallfac[3,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,1](x, m), n >= 0.
%D A290596 Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.
%H A290596 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 A290596 Wolfdieter Lang, <a href="/A290597/a290597.pdf">Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.</a>
%F A290596 T(n, m) = L[3,1](n,m) = Sum_{k=m..n} A286718(n, k)*A111577(k+1, m+1), 0 <= m <= n.
%F A290596 E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
%F A290596 (1 - 3*t)^(-2/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
%F A290596 E.g.f. of column m: (1 - 3*t)^(-2/3)*(t/(1 - 3*t))^m/m!, m >= 0.
%F A290596 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 A290596 Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(3*n - 2)*T(n-1, m) - 3*(n-1)*(3*n - 4)*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 A290596 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 A290596 General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
%F A290596 R(n, x) = [(2 + x)*1 + 6*(1 + x)*D_x + 3^2*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
%F A290596 Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(2 + 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 A290596 The triangle T(n, m) begins:
%e A290596 n\m 0 1 2 3 4 5 6 7 8 ...
%e A290596 0: 1
%e A290596 1: 2 1
%e A290596 2: 10 10 1
%e A290596 3: 80 120 24 1
%e A290596 4: 880 1760 528 44 1
%e A290596 5: 12320 30800 12320 1540 70 1
%e A290596 6: 209440 628320 314160 52360 3570 102 1
%e A290596 7: 4188800 14660800 8796480 1832600 166600 7140 140 1
%e A290596 8: 96342400 385369600 269758720 67439680 7663600 437920 12880 184 1
%e A290596 ...
%e A290596 n = 9: 2504902400 11272060800 9017648640 2630147520 358656480 25618320 1004640 21528 234 1,
%e A290596 n = 10: 72642169600 363210848000 326889763200 108963254400 17335063200 1485862560 72836400 2081040 33930 290 1.
%e A290596 ...
%e A290596 Recurrence from a-sequence: T(4, 2) = 2*T(3, 1) + 3*4*T(3, 2) = 2*120 + 12*24 = 528.
%e A290596 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*80 + 1*120 - (10/3)*24 + 20*1) = 880.
%e A290596 Four term recurrence: T(4, 2) = T(3, 1) + 2*10*T(3, 2) - 3*3*8*T(2, 2) = 120 + 20*24 - 72*1 = 528.
%e A290596 Meixner type identity for n = 2: (D_x - 3*(D_x)^2)*(10 + 10*x + x^2 ) = (10 + 2*x) - 3*2 = 2*(2 + x).
%e A290596 Sheffer recurrence for R(3, x): [(2 + x) + 6*(1 + x)*D_x + 9*x*(D_x)^2] (10 + 10*x + x^2) = (2 + x)*(10 + 10*x + x^2) + 6*(1 + x)*(10 +2*x) + 9*2*x = 80 + 120*x + 24*x^2 + x^3 = R(3, x).
%e A290596 Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*8/2)*(1*24/3! + 3*1/2!) = 528.
%Y A290596 Cf. A008544 (column m=0), A038500, A111577, A271703 L[1,0], A286718, A286724 L[2,1], A290597, A290598 L[3,2].
%K A290596 nonn,easy,tabl
%O A290596 0,2
%A A290596 _Wolfdieter Lang_, Sep 13 2017