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
Examples
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.
Programs
-
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;
-
PARI
M(n) = prod(i=1, #factor(n!)~, prime(i)^sum(k=0, #binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k)))) \\ from A053657 rows_upto(n) = my(v1, v2); v1 = vector(n, i, 0); v2 = vector(n+1, i, 0); v2[1] = 1; for(i=1, n, v1[i] = (i+x)*(i+x-1)/2*v2[i]; for(j=1, i-1, v1[j] *= (i-j)*(i+x)/(i-j+2)); v2[i+1] = vecsum(v1)/i); v2 = vector(n+1, i, M(i)*Vecrev(v2[i])) \\ Mikhail Kurkov, Aug 27 2025