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A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.

%I A294498 #20 Jan 06 2023 15:45:33
%S A294498 1,1,0,1,1,0,1,2,2,0,1,3,6,5,0,1,4,12,22,14,0,1,5,20,57,92,42,0,1,6,
%T A294498 30,116,306,424,132,0,1,7,42,205,752,1806,2108,429,0,1,8,56,330,1550,
%U A294498 5328,11508,11134,1430,0,1,9,72,497,2844,12730,40632,78147,61748,4862,0
%N A294498 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.
%C A294498 A(n,k) is the k-fold exponential convolution of Catalan numbers with themselves, evaluated at n.
%H A294498 Alois P. Heinz, <a href="/A294498/b294498.txt">Antidiagonals for n = 0..150, flattened</a>
%F A294498 E.g.f. of column k: exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.
%e A294498 E.g.f. of column k: A_k(x) = 1 + k*x/1! +  k*(k + 1)*x^2/2! + k*(k^2 + 3*k + 1)*x^3/3! + k^2*(k^2 + 6*k + 7)*x^4/4! + k*(k^4 + 10*k^3 + 25*k^2 + 10*k - 4)*x^5/5! + ...
%e A294498 Square array begins:
%e A294498   1,   1,    1,     1,     1,      1,  ...
%e A294498   0,   1,    2,     3,     4,      5,  ...
%e A294498   0,   2,    6,    12,    20,     30,  ...
%e A294498   0,   5,   22,    57,   116,    205,  ...
%e A294498   0,  14,   92,   306,   752,   1550,  ...
%e A294498   0,  42,  424,  1806,  5328,  12730,  ...
%p A294498 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A294498 A:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, C(n),
%p A294498       (h-> add(binomial(n, j)*A(j, h)*A(n-j, k-h), j=0..n))(iquo(k, 2))))
%p A294498     end:
%p A294498 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Jan 06 2023
%t A294498 Table[Function[k, n! SeriesCoefficient[Exp[2 k x] (BesselI[0, 2 x] - BesselI[1, 2 x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y A294498 Columns k=0..3 give A000007, A000108, A014330, A014333.
%Y A294498 Rows n=0..2 give  A000012, A001477, A002378.
%Y A294498 Main diagonal gives A294511.
%Y A294498 Cf. A009766, A033184.
%K A294498 nonn,tabl
%O A294498 0,8
%A A294498 _Ilya Gutkovskiy_, Nov 01 2017