A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

1, 0, 1, 0, 4, 3, 0, 46, 60, 15, 0, 1114, 1848, 840, 105, 0, 46246, 88770, 54180, 12600, 945, 0, 2933074, 6235548, 4574130, 1469160, 207900, 10395, 0, 263817646, 605964450, 505915410, 199849650, 39729690, 3783780, 135135

Offset: 0

Author

Peter Luschny, Jul 08 2019

Keywords

• nonn
• tabl

Examples

			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]
		

Crossrefs

Row sums A094088. Alternating row sums A153881 starting at 0.

Main diagonal A001147. Associated set partitions A241171.

A129062 (m=1, associated with A131689), this sequence (m=2), A326587 (m=3, associated with A278073), A326585 (m=4, associated with A278074).

Programs

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 *)

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))

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).