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

0,6

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

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.

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;

Crossrefs

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

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

Sequence in context: A334823 A279031 A304336 * A350212 A256311 A022695

Adjacent sequences: A287312 A287313 A287314 * A287316 A287317 A287318

Keyword

nonn,tabl

Author

Peter Luschny, May 29 2017

Status

approved