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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.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 29 2017
STATUS
approved