A072024 Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 7, 1, 1, 0, 1, 11, 13, 13, 1, 1, 0, 1, 21, 55, 25, 21, 1, 1, 0, 1, 43, 133, 181, 41, 31, 1, 1, 0, 1, 85, 463, 481, 461, 61, 43, 1, 1, 0, 1, 171, 1261, 2653, 1281, 991, 85, 57, 1, 1, 0, 1, 341, 4039, 8425, 10501, 2821, 1891, 113, 73, 1, 1, 0

Offset: 0

Author

Henry Bottomley, Jun 06 2002

Keywords

• nonn
• tabl

Comments

Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - Philippe Deléham, Nov 24 2013

Examples

			Rows start:
0 1 1  1   1    1     1      1       1        1 ...
0 1 1  3   5   11    21     43      85      171 ...
0 1 1  7  13   55   133    463    1261     4039 ...
0 1 1 13  25  181   481   2653    8425    40261 ...
0 1 1 21  41  461  1281  10501   36121   246141 ...
0 1 1 31  61  991  2821  32551  117181  1093711 ...
0 1 1 43  85 1891  5461  84883  314245  3879331 ...
0 1 1 57 113 3305  9633 194713  734161 11638089 ...
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1275

Crossrefs

Rows include A057427, A001045, A015441, A053404, A053428, A053430, A065874, etc. Columns include A000004, A000012, A000012, A002061, A001844, A072025, etc.

Cf. A081297.

Programs

Magma

[((k+1)^(n-k) - (-k)^(n-k))/(2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020

Maple

seq(seq( ((k+1)^(n-k) - (-k)^(n-k))/(2*k+1), k=0..n), n=0..12); # G. C. Greubel, Jan 27 2020

Mathematica

T[n_, k_]:= ((n + 1)^k - (-n)^k)/(2n + 1); Flatten[Join[{0}, Table[T[k, n- k], {n, 1, 15}, {k, 0, n}]]] (* Indranil Ghosh, Mar 27 2017 *)

PARI

for(n=0, 10, for(k=0, 9, print1(((n+1)^k-(-n)^k)/(2*n+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017

Sage

def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)
[[T(k,n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

Formula

T(n, k) = T(n, k-1) + n*(n+1)*T(n, k-2) = A060959(A002378(n), k).

T(k, 2n) = (2n+1)*A047969(n, k+1).