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%I #9 Sep 20 2017 20:00:26
%S 0,0,1,0,1,0,0,1,2,3,0,1,4,6,0,0,1,6,15,16,10,0,1,8,30,56,45,0,0,1,10,
%T 51,144,210,126,35,0,1,12,78,304,685,792,357,0,0,1,14,111,560,1770,
%U 3258,3003,1016,126,0,1,16,150,936,3885,10224,15533,11440,2907,0,0,1,18,195,1456,7570,26550,58947,74280,43758,8350,462
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*BesselI(1,2*x).
%C A(n,k) is the k-th binomial transform of A138364 evaluated at n.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F E.g.f. of column k: exp(k*x)*BesselI(1,2*x).
%e E.g.f. of column k: A_k(x) = x/1! + 2*k*x^2/2! + 3*(k^2 + 1)*x^3/3! + 4*k*(k^2 + 3)*x^4/4! + 5*(k^4 + 6*k^2 + 2)*x^5/5! + ...
%e Square array begins:
%e 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, 10, ...
%e 3, 6, 15, 30, 51, 78, ...
%e 0, 16, 56, 144, 304, 560, ...
%e 10, 45, 210, 685, 1770, 3885, ...
%t Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[1, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0..3 give A138364, A005717, A001791, A026376.
%Y Main diagonal gives A292629.
%Y Cf. A292627.
%K nonn,tabl
%O 0,9
%A _Ilya Gutkovskiy_, Sep 20 2017