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

%I A072024 #28 Sep 08 2022 08:45:06
%S A072024 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 A072024 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 A072024 171,1261,2653,1281,991,85,57,1,1,0,1,341,4039,8425,10501,2821,1891,113,73,1,1,0
%N A072024 Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).
%C A072024 Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - _Philippe Deléham_, Nov 24 2013
%H A072024 Andrew Howroyd, <a href="/A072024/b072024.txt">Table of n, a(n) for n = 0..1275</a>
%F A072024 T(n, k) = T(n, k-1) + n*(n+1)*T(n, k-2) = A060959(A002378(n), k).
%F A072024 T(k, 2n) = (2n+1)*A047969(n, k+1).
%e A072024 Rows start:
%e A072024 0 1 1  1   1    1     1      1       1        1 ...
%e A072024 0 1 1  3   5   11    21     43      85      171 ...
%e A072024 0 1 1  7  13   55   133    463    1261     4039 ...
%e A072024 0 1 1 13  25  181   481   2653    8425    40261 ...
%e A072024 0 1 1 21  41  461  1281  10501   36121   246141 ...
%e A072024 0 1 1 31  61  991  2821  32551  117181  1093711 ...
%e A072024 0 1 1 43  85 1891  5461  84883  314245  3879331 ...
%e A072024 0 1 1 57 113 3305  9633 194713  734161 11638089 ...
%e A072024 ...
%p A072024 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 A072024 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 A072024 (PARI)
%o A072024 for(n=0, 10, for(k=0, 9, print1(((n+1)^k-(-n)^k)/(2*n+1), ", "); ); print(); ) \\ _Andrew Howroyd_, Mar 26 2017
%o A072024 (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 A072024 (Sage)
%o A072024 def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)
%o A072024 [[T(k,n-k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 27 2020
%Y A072024 Rows include A057427, A001045, A015441, A053404, A053428, A053430, A065874, etc. Columns include A000004, A000012, A000012, A002061, A001844, A072025, etc.
%Y A072024 Cf. A081297.
%K A072024 nonn,tabl
%O A072024 0,12
%A A072024 _Henry Bottomley_, Jun 06 2002