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, 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

Links

Table of n, a(n) for n=0..48.

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

Cf. A053657, A163972, A008517, A201637, A340556, A341110 (row sums), A340556.

Sequence in context: A160479 A085222 A085221 * A128536 A202318 A251128

Adjacent sequences: A341108 A341109 A341110 * A341112 A341113 A341114

Keyword

nonn,tabf

Author

Peter Luschny, Feb 05 2021

Status

approved