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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.
2
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, 30, 116, 306, 424, 132, 0, 1, 7, 42, 205, 752, 1806, 2108, 429, 0, 1, 8, 56, 330, 1550, 5328, 11508, 11134, 1430, 0, 1, 9, 72, 497, 2844, 12730, 40632, 78147, 61748, 4862, 0
OFFSET
0,8
COMMENTS
A(n,k) is the k-fold exponential convolution of Catalan numbers with themselves, evaluated at n.
FORMULA
E.g.f. of column k: exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.
EXAMPLE
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! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 6, 12, 20, 30, ...
0, 5, 22, 57, 116, 205, ...
0, 14, 92, 306, 752, 1550, ...
0, 42, 424, 1806, 5328, 12730, ...
MAPLE
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, C(n),
(h-> add(binomial(n, j)*A(j, h)*A(n-j, k-h), j=0..n))(iquo(k, 2))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Jan 06 2023
MATHEMATICA
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
CROSSREFS
Columns k=0..3 give A000007, A000108, A014330, A014333.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A294511.
Sequence in context: A085388 A351339 A361432 * A292860 A265609 A362125
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Nov 01 2017
STATUS
approved