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A341111
T(n, k) = [x^k] M(n)*Sum_{k=0..n} E2(n, k)*binomial(-x + n - k, 2*n), where E2 are the second-order Eulerian numbers A340556 and M(n) are the Minkowski numbers A053657. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= 2*n+1.
1
1, 0, 1, 1, 0, 10, 21, 14, 3, 0, 36, 96, 97, 47, 11, 1, 0, 12048, 36740, 45420, 29855, 11352, 2510, 300, 15, 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3, 0, 109941120, 392583744, 603023624, 531477324, 300731214, 115291701, 30675678, 5682033, 719866, 59535, 2898, 63
OFFSET
0,6
EXAMPLE
Triangle starts:
[0] 1;
[1] 0, 1, 1;
[2] 0, 10, 21, 14, 3;
[3] 0, 36, 96, 97, 47, 11, 1;
[4] 0, 12048, 36740, 45420, 29855, 11352, 2510, 300, 15;
[5] 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3.
MAPLE
E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x))]:
mser := series((y/(exp(y)-1))^x, y, 29): m := n -> denom(coeff(mser, y, n)):
poly := n -> expand(m(n)*add(E2(n, k)*binomial(-x+n-k, 2*n), k = 0..n)):
for n from 0 to 6 do CoeffList(poly(n)) od;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Feb 05 2021
STATUS
approved