%I #28 Sep 08 2022 08:45:06

%S 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,

%T 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,

%U 171,1261,2653,1281,991,85,57,1,1,0,1,341,4039,8425,10501,2821,1891,113,73,1,1,0

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

%C Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - _Philippe Deléham_, Nov 24 2013

%H Andrew Howroyd, <a href="/A072024/b072024.txt">Table of n, a(n) for n = 0..1275</a>

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

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

%e Rows start:

%e 0 1 1 1 1 1 1 1 1 1 ...

%e 0 1 1 3 5 11 21 43 85 171 ...

%e 0 1 1 7 13 55 133 463 1261 4039 ...

%e 0 1 1 13 25 181 481 2653 8425 40261 ...

%e 0 1 1 21 41 461 1281 10501 36121 246141 ...

%e 0 1 1 31 61 991 2821 32551 117181 1093711 ...

%e 0 1 1 43 85 1891 5461 84883 314245 3879331 ...

%e 0 1 1 57 113 3305 9633 194713 734161 11638089 ...

%e ...

%p seq(seq( ((k+1)^(n-k) - (-k)^(n-k))/(2*k+1), k=0..n), n=0..12); # _G. C. Greubel_, Jan 27 2020

%t 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 *)

%o (PARI)

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

%o (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

%o (Sage)

%o def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)

%o [[T(k,n-k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 27 2020

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

%Y Cf. A081297.

%K nonn,tabl

%O 0,12

%A _Henry Bottomley_, Jun 06 2002