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.

1, 0, 1, 0, 1, 3, 0, 1, 16, 19, 0, 1, 65, 299, 211, 0, 1, 246, 3156, 7346, 3651, 0, 1, 917, 28722, 160322, 237517, 90921, 0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513, 0, 1, 12861, 2041965, 46261609, 288196659, 632274183, 520507423, 136407699

Offset: 0

Author

Peter Luschny, May 29 2017

Keywords

• nonn
• tabl

Examples

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1,    3]
3: [0, 1,   16,     19]
4: [0, 1,   65,    299,     211]
5: [0, 1,  246,   3156,    7346,    3651]
6: [0, 1,  917,  28722,  160322,  237517,   90921]
7: [0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513]
...
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.
		

Crossrefs

T(n,n) = A000275(n).

Cf. A192721 (variant), A001044, A287314, A287316.

Programs

Maple

Delta := proc(a, n) local del, A, u;
A := [seq(a(j), j=0..n+1)]; del := (a, k) -> `if`(k=0, a(0), a(k)-a(k-1));
for u from 0 to n do A := [seq(del(k -> A[k+1], j), j=0..n)] od end:
A287315_row := n -> Delta(A287314_poly(n), n):
for n from 0 to 7 do A287315_row(n) od;
A287315_eulerian := (n,x) -> add(A287315_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287315_eulerian(n,x) od;

Formula

Sum_{k=0..n} T(n,k) = A001044(n).