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A292628 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).

%I A292628 #9 Sep 20 2017 20:00:26
%S A292628 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 A292628 51,144,210,126,35,0,1,12,78,304,685,792,357,0,0,1,14,111,560,1770,
%U A292628 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 A292628 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 A292628 A(n,k) is the k-th binomial transform of A138364 evaluated at n.
%H A292628 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292628 E.g.f. of column k: exp(k*x)*BesselI(1,2*x).
%e A292628 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 A292628 Square array begins:
%e A292628    0,   0,    0,    0,     0,     0,  ...
%e A292628    1,   1,    1,    1,     1,     1,  ...
%e A292628    0,   2,    4,    6,     8,    10,  ...
%e A292628    3,   6,   15,   30,    51,    78,  ...
%e A292628    0,  16,   56,  144,   304,   560,  ...
%e A292628   10,  45,  210,  685,  1770,  3885,  ...
%t A292628 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 A292628 Columns k=0..3 give A138364, A005717, A001791, A026376.
%Y A292628 Main diagonal gives A292629.
%Y A292628 Cf. A292627.
%K A292628 nonn,tabl
%O A292628 0,9
%A A292628 _Ilya Gutkovskiy_, Sep 20 2017