A326477 - OEIS

%I #21 Mar 06 2020 09:28:53

%S 1,0,1,0,4,3,0,46,60,15,0,1114,1848,840,105,0,46246,88770,54180,12600,

%T 945,0,2933074,6235548,4574130,1469160,207900,10395,0,263817646,

%U 605964450,505915410,199849650,39729690,3783780,135135

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

%F 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).

%e Triangle starts:

%e [0] [1]

%e [1] [0, 1]

%e [2] [0, 4, 3]

%e [3] [0, 46, 60, 15]

%e [4] [0, 1114, 1848, 840, 105]

%e [5] [0, 46246, 88770, 54180, 12600, 945]

%e [6] [0, 2933074, 6235548, 4574130, 1469160, 207900, 10395]

%p CL := f -> PolynomialTools:-CoefficientList(f, x):

%p FL := s -> ListTools:-Flatten(s, 1):

%p StirPochConv := proc(m, n) local P, L; P := proc(m, n) option remember;

%p `if`(n = 0, 1, add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n)) end:

%p L := CL(P(m, n)); CL(expand(add(L[k+1]*pochhammer(x,k)/k!, k=0..n))) end:

%p FL([seq(StirPochConv(2,n), n = 0..7)]);

%t 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 T[m_][n_] := CoefficientList[P[m, n], x].Table[Pochhammer[x, k]/k!, {k, 0, n}] // CoefficientList[#, x]&;

%t Table[T[2][n], {n, 0, 7}] // Flatten (* _Jean-François Alcover_, Jul 21 2019 *)

%o (Sage)

%o def StirPochConv(m, n):

%o     z = var('z'); R = ZZ[x]

%o     F = [i/m for i in (1..m-1)]

%o     H = hypergeometric([], F, (z/m)^m)

%o     P = R(factorial(m*n)*taylor(exp(x*(H-1)), z, 0, m*n + 1).coefficient(z, m*n))

%o     L = P.list()

%o     S = sum(L[k]*rising_factorial(x,k) for k in (0..n))

%o     return expand(S).list()

%o for n in (0..6): print(StirPochConv(2, n))

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

%Y Main diagonal A001147. Associated set partitions A241171.

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

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Jul 08 2019