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A064307 Triangle of coefficients of certain numerator polynomials N(n,x).

%I A064307 #9 Jan 07 2025 20:26:44
%S A064307 1,1,0,1,2,1,1,10,17,2,1,37,181,111,6,1,126,1530,2624,741,18,1,422,
%T A064307 11607,43940,34063,4950,57,1,1422,83823,616894,1013799,412698,33337,
%U A064307 186,1,4853,593203,7846573,23794925
%N A064307 Triangle of coefficients of certain numerator polynomials N(n,x).
%C A064307 The g.f. for the sequence in the subdiagonal d>=1 (main diagonal: d=0) of triangle A064094 is N(d,x)/(1-x)^d.
%C A064307 Row sums give A001761(n+1). Main diagonal gives A000957(n+1), n >= 0.
%F A064307 a(n, m) = [x^m]N(n, x); N(n, x) = (1-x)^(n-1) + Sum_{k=1..n-1} A064308(n-1, k)*k!*x^k*(1-x)^(n-1-k) for n >= 2; N(1, x) = 1 = N(2, x).
%e A064307 Triangle begins:
%e A064307   1;
%e A064307   1, 0;
%e A064307   1, 2,  1;       N(3,x) = 1+2*x+x^2 = (1+x)^2.
%e A064307   1, 10, 17,  2;
%e A064307   1, 37, 181, 111, 6;
%e A064307   ...
%Y A064307 Cf. A000957, A001761, A064094.
%K A064307 nonn,tabl
%O A064307 1,5
%A A064307 _Wolfdieter Lang_, Sep 13 2001