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A287315 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.

%I A287315 #13 Jan 10 2018 20:33:26
%S A287315 1,0,1,0,1,3,0,1,16,19,0,1,65,299,211,0,1,246,3156,7346,3651,0,1,917,
%T A287315 28722,160322,237517,90921,0,1,3424,245407,2864912,9302567,9903776,
%U A287315 3081513,0,1,12861,2041965,46261609,288196659,632274183,520507423,136407699
%N A287315 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.
%F A287315 Sum_{k=0..n} T(n,k) = A001044(n).
%e A287315 Triangle starts:
%e A287315 0: [1]
%e A287315 1: [0, 1]
%e A287315 2: [0, 1,    3]
%e A287315 3: [0, 1,   16,     19]
%e A287315 4: [0, 1,   65,    299,     211]
%e A287315 5: [0, 1,  246,   3156,    7346,    3651]
%e A287315 6: [0, 1,  917,  28722,  160322,  237517,   90921]
%e A287315 7: [0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513]
%e A287315 ...
%e A287315 Let q4(x) = (x + 65*x^2 + 299*x^3 + 211*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 give A169712, which is column 4 of A287316.
%p A287315 Delta := proc(a, n) local del, A, u;
%p A287315 A := [seq(a(j), j=0..n+1)]; del := (a, k) -> `if`(k=0, a(0), a(k)-a(k-1));
%p A287315 for u from 0 to n do A := [seq(del(k -> A[k+1], j), j=0..n)] od end:
%p A287315 A287315_row := n -> Delta(A287314_poly(n), n):
%p A287315 for n from 0 to 7 do A287315_row(n) od;
%p A287315 A287315_eulerian := (n,x) -> add(A287315_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
%p A287315 for n from 0 to 4 do A287315_eulerian(n,x) od;
%Y A287315 T(n,n) = A000275(n).
%Y A287315 Cf. A192721 (variant), A001044, A287314, A287316.
%K A287315 nonn,tabl
%O A287315 0,6
%A A287315 _Peter Luschny_, May 29 2017