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.

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

Author

Ilya Gutkovskiy, Nov 01 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the k-fold exponential convolution of Catalan numbers with themselves, evaluated at n.

Examples

			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,  ...
		

Links

• Alois P. Heinz, Antidiagonals for n = 0..150, flattened

Crossrefs

Columns k=0..3 give A000007, A000108, A014330, A014333.

Rows n=0..2 give A000012, A001477, A002378.

Main diagonal gives A294511.

Cf. A009766, A033184.

Programs

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

Formula

E.g.f. of column k: exp(2*k*x)*(BesselI(0,2*x) - BesselI(1,2*x))^k.