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
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
Programs
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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
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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
Comments