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A292630 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(0,2*x) + BesselI(1,2*x)).

%I A292630 #10 Sep 20 2017 20:03:06
%S A292630 1,1,1,1,2,2,1,3,5,3,1,4,10,13,6,1,5,17,35,35,10,1,6,26,75,126,96,20,
%T A292630 1,7,37,139,339,462,267,35,1,8,50,233,758,1558,1716,750,70,1,9,65,363,
%U A292630 1491,4194,7247,6435,2123,126,1,10,82,535,2670,9660,23460,34016,24310,6046,252,1,11,101,755,4451,19846,63195,132339,160795,92378,17303,462
%N A292630 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(0,2*x) + BesselI(1,2*x)).
%C A292630 A(n,k) is the k-th binomial transform of A001405 evaluated at n.
%H A292630 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292630 E.g.f. of column k: exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).
%e A292630 E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 2)*x^2/2! +  (k^3 + 3*k^2 + 6*k + 3)*x^3/3! + (k^4 + 4*k^3 + 12*k^2 + 12*k + 6)*x^4/4! + ...
%e A292630 Square array begins:
%e A292630    1,   1,    1,     1,     1,     1,  ...
%e A292630    1,   2,    3,     4,     5,     6,  ...
%e A292630    2,   5,   10,    17,    26,    37,  ...
%e A292630    3,  13,   35,    75,   139,   233,  ...
%e A292630    6,  35,  126,   339,   758,  1491,  ...
%e A292630   10,  96,  462,  1558,  4194,  9660,  ...
%p A292630 [seq(seq((k)!*add((m-j)^(j-i)/floor(i/2)!/ceil(i/2)!/(j-i)!,i=0..j),j=0..m), m=0..20)]; # _Robert Israel_, Sep 20 2017
%t A292630 Table[Function[k, n! SeriesCoefficient[Exp[k x] (BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y A292630 Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.
%Y A292630 Main diagonal gives A292631.
%K A292630 nonn,tabl
%O A292630 0,5
%A A292630 _Ilya Gutkovskiy_, Sep 20 2017