A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 13, 13, 11, 1, 1, 1, 21, 25, 55, 21, 1, 1, 1, 31, 41, 181, 133, 43, 1, 1, 1, 43, 61, 461, 481, 463, 85, 1, 1, 1, 57, 85, 991, 1281, 2653, 1261, 171, 1, 1, 1, 73, 113, 1891, 2821, 10501, 8425, 4039, 341, 1, 1, 1, 91, 145, 3305

Offset

0,9

Comments

Square array of solutions of a family of recurrences.

Rows of the array give solutions to the recurrences a(n)=a(n-1)+k(k-1)a(n-2), a(0)=a(1)=1.

Subarray of array in A072024. - Philippe Deléham, Nov 24 2013

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Formula

T(k, n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

Rows of the array have g.f. 1/((1+kx)(1-(k+1)x)).

Example

Rows begin
  1, 1,  1,  1,   1,    1, ...
  1, 1,  3,  5,  11,   21, ...
  1, 1,  7, 13,  55,  133, ...
  1, 1, 13, 25, 181,  481, ...
  1, 1, 21, 41, 461, 1281, ...

Mathematica

T[n_, k_]:=((n + 1)^(k + 1) - (-n)^(k + 1)) / (2n + 1); Flatten[Table[T[n - k, k], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 27 2017 *)

Prog

(PARI)
for(k=0, 10, for(n=0, 9, print1(((k+1)^(n+1)-(-k)^(n+1))/(2*k+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017
(Python)
def T(n, k): return ((n + 1)**(k + 1) - (-n)**(k + 1)) // (2*n + 1)
for n in range(11):
    print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Mar 27 2017

Crossrefs

Cf. A059259, A072024.

Rows include A001045, A015441, A053404, A053428, A053430.

Columns include A002061, A001844, A072025.

Diagonals include A081298, A081299, A081300, A081301, A081302.

Sequence in context: A243473 A325969 A325826 * A110180 A328718 A362897

Adjacent sequences: A081294 A081295 A081296 * A081298 A081299 A081300

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 17 2003

Extensions

Name clarified by Andrew Howroyd, Mar 27 2017

Status

approved