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
KEYWORD
AUTHOR
Paul Barry, Mar 17 2003
EXTENSIONS
Name clarified by Andrew Howroyd, Mar 27 2017
STATUS
approved