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

Author

Paul Barry, Mar 17 2003

Keywords

• easy
• nonn
• tabl

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

Examples

			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, ...
		

Links

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

Crossrefs

Cf. A059259, A072024.

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

Columns include A002061, A001844, A072025.

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

Programs

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 *)

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

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)).

Extensions

Name clarified by Andrew Howroyd, Mar 27 2017