OFFSET
0,2
COMMENTS
For the general L[d,a] triangles see A286724, also for references.
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.
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.
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).
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).
fallfac[3,2](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[3,2](x, m), n >= 0.
REFERENCES
Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.
LINKS
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
FORMULA
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 3*t)^(-4/3)*exp(x*t/(1 - 3*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 3*t)^(-4/3)*(t/(1 - 3*t))^m/m!, m >= 0.
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.
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.
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.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
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.
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.
EXAMPLE
The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7 8 ...
0: 1
1: 4 1
2: 28 14 1
3: 280 210 30 1
4: 3640 3640 780 52 1
5: 58240 72800 20800 2080 80 1
6: 1106560 1659840 592800 79040 4560 114 1
7: 24344320 42602560 18258240 3043040 234080 8778 154 1
8: 608608000 1217216000 608608000 121721600 11704000 585200 15400 200 1
...
n = 9: 17041024000 38342304000 21909888000 5112307200 589881600 36867600 1293600 25200 252 1,
n = 10: 528271744000 1320679360000 849008160000 226402176000 30477216000 2285791200 100254000 2604000 39060 310 1.
...
Recurrence from a-sequence: T(4, 2) = (4/2)*T(3, 1) + 3*4*T(3, 2) = 2*210 + 12*30 = 780.
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.
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.
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).
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).
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.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 13 2017
STATUS
approved