OFFSET
0,5
FORMULA
For m >= 1 let P(m,0) = 1 and P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x for n > 0. Then T_{m}(n, k) = Sum_{k=0..n} ([x^k]P(m, n))*rf(x,k)/k! where rf(x,k) are the rising factorial powers. T(n, k) = T_{2}(n, k).
EXAMPLE
Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 4, 3]
[3] [0, 46, 60, 15]
[4] [0, 1114, 1848, 840, 105]
[5] [0, 46246, 88770, 54180, 12600, 945]
[6] [0, 2933074, 6235548, 4574130, 1469160, 207900, 10395]
MAPLE
CL := f -> PolynomialTools:-CoefficientList(f, x):
FL := s -> ListTools:-Flatten(s, 1):
StirPochConv := proc(m, n) local P, L; P := proc(m, n) option remember;
`if`(n = 0, 1, add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n)) end:
L := CL(P(m, n)); CL(expand(add(L[k+1]*pochhammer(x, k)/k!, k=0..n))) end:
FL([seq(StirPochConv(2, n), n = 0..7)]);
MATHEMATICA
P[_, 0] = 1; P[m_, n_] := P[m, n] = Sum[Binomial[m*n, m*k]*P[m, n-k]*x, {k, 1, n}] // Expand;
T[m_][n_] := CoefficientList[P[m, n], x].Table[Pochhammer[x, k]/k!, {k, 0, n}] // CoefficientList[#, x]&;
Table[T[2][n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)
PROG
(Sage)
def StirPochConv(m, n):
z = var('z'); R = ZZ[x]
F = [i/m for i in (1..m-1)]
H = hypergeometric([], F, (z/m)^m)
P = R(factorial(m*n)*taylor(exp(x*(H-1)), z, 0, m*n + 1).coefficient(z, m*n))
L = P.list()
S = sum(L[k]*rising_factorial(x, k) for k in (0..n))
return expand(S).list()
for n in (0..6): print(StirPochConv(2, n))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jul 08 2019
STATUS
approved