cp's OEIS Frontend

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.

%I A341111 #17 Sep 01 2025 21:33:49
%S A341111 1,0,1,1,0,10,21,14,3,0,36,96,97,47,11,1,0,12048,36740,45420,29855,
%T A341111 11352,2510,300,15,0,91200,304480,427348,334620,162255,50787,10302,
%U A341111 1310,95,3,0,109941120,392583744,603023624,531477324,300731214,115291701,30675678,5682033,719866,59535,2898,63
%N 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.
%e A341111 Triangle starts:
%e A341111 [0] 1;
%e A341111 [1] 0, 1,     1;
%e A341111 [2] 0, 10,    21,     14,     3;
%e A341111 [3] 0, 36,    96,     97,     47,     11,     1;
%e A341111 [4] 0, 12048, 36740,  45420,  29855,  11352,  2510,  300,   15;
%e A341111 [5] 0, 91200, 304480, 427348, 334620, 162255, 50787, 10302, 1310, 95, 3.
%p A341111 E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
%p A341111 CoeffList := p -> [op(PolynomialTools:-CoefficientList(p, x))]:
%p A341111 mser := series((y/(exp(y)-1))^x, y, 29): m := n -> denom(coeff(mser, y, n)):
%p A341111 poly := n -> expand(m(n)*add(E2(n, k)*binomial(-x+n-k, 2*n), k = 0..n)):
%p A341111 for n from 0 to 6 do CoeffList(poly(n)) od;
%o A341111 (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
%o A341111 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
%Y A341111 Cf. A053657, A163972, A008517, A201637, A340556, A341110 (row sums), A340556.
%K A341111 nonn,tabf,changed
%O A341111 0,6
%A A341111 _Peter Luschny_, Feb 05 2021