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).

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, 51, 144, 210, 126, 35, 0, 1, 12, 78, 304, 685, 792, 357, 0, 0, 1, 14, 111, 560, 1770, 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

Offset: 0

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the k-th binomial transform of A138364 evaluated at n.

Examples

			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! + ...
Square array begins:
   0,   0,    0,    0,     0,     0,  ...
   1,   1,    1,    1,     1,     1,  ...
   0,   2,    4,    6,     8,    10,  ...
   3,   6,   15,   30,    51,    78,  ...
   0,  16,   56,  144,   304,   560,  ...
  10,  45,  210,  685,  1770,  3885,  ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Columns k=0..3 give A138364, A005717, A001791, A026376.

Main diagonal gives A292629.

Cf. A292627.

Programs

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[k x] BesselI[1, 2 x], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

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