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A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

%I A089944 #22 Jan 13 2025 16:30:27
%S A089944 1,2,1,3,3,1,4,8,4,1,5,20,15,5,1,6,48,54,24,6,1,7,112,189,112,35,7,1,
%T A089944 8,256,648,512,200,48,8,1,9,576,2187,2304,1125,324,63,9,1,10,1280,
%U A089944 7290,10240,6250,2160,490,80,10,1,11,2816,24057,45056,34375,14256,3773,704,99,11,1
%N A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.
%C A089944 The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.
%H A089944 Paolo Xausa, <a href="/A089944/b089944.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals).
%F A089944 T(n,k) = (k+n+1)*(n+1)^(k-1).
%F A089944 E.g.f.: (1+x)*exp(x)/(1-y*exp(x)).
%e A089944 Rows begin:
%e A089944   {1, 2, 3, 4, 5, 6, 7,..},
%e A089944   {1, 3, 8, 20, 48, 112, 256,..},
%e A089944   {1, 4, 15, 54, 189, 648, 2187,..},
%e A089944   {1, 5, 24, 112, 512, 2304, 10240,..},
%e A089944   {1, 6, 35, 200, 1125, 6250, 34375,..},
%e A089944   {1, 7, 48, 324, 2160, 14256, 93312,..},
%e A089944   {1, 8, 63, 490, 3773, 28812, 218491,..},..
%t A089944 A089944[n_, k_] := (k + n + 1)*(n + 1)^(k - 1);
%t A089944 Table[A089944[k, n - k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 13 2025 *)
%o A089944 (PARI) T(n,k)=if(n<0 || k<0,0,(k+n+1)*(n+1)^(k-1))
%Y A089944 Cf. A089945, A089946.
%Y A089944 Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.
%Y A089944 Columns : A000012, A000027, A005563.
%K A089944 nonn,tabl,easy
%O A089944 0,2
%A A089944 _Paul D. Hanna_, Nov 23 2003