|
%I #12 Feb 17 2021 13:07:39
%S 1,0,1,1,0,10,21,14,3,0,36,96,97,47,11,1,0,12048,36740,45420,29855,
%T 11352,2510,300,15,0,91200,304480,427348,334620,162255,50787,10302,
%U 1310,95,3,0,109941120,392583744,603023624,531477324,300731214,115291701,30675678,5682033,719866,59535,2898,63
%N 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.
%e Triangle starts:
%e [0] 1;
%e [1] 0, 1, 1;
%e [2] 0, 10, 21, 14, 3;
%e [3] 0, 36, 96, 97, 47, 11, 1;
%e [4] 0, 12048, 36740, 45420, 29855, 11352, 2510, 300, 15;
%e [5] 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3.
%p E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
%p CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x))]:
%p mser := series((y/(exp(y)-1))^x, y, 29): m := n -> denom(coeff(mser, y, n)):
%p poly := n -> expand(m(n)*add(E2(n, k)*binomial(-x+n-k, 2*n), k = 0..n)):
%p for n from 0 to 6 do CoeffList(poly(n)) od;
%Y Cf. A053657, A163972, A008517, A201637, A340556, A341110 (row sums), A340556.
%K nonn,tabf
%O 0,6
%A _Peter Luschny_, Feb 05 2021
|
