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)).

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, 1, 7, 37, 139, 339, 462, 267, 35, 1, 8, 50, 233, 758, 1558, 1716, 750, 70, 1, 9, 65, 363, 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

Offset: 0

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the k-th binomial transform of A001405 evaluated at n.

Examples

			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! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   2,   5,   10,    17,    26,    37,  ...
   3,  13,   35,    75,   139,   233,  ...
   6,  35,  126,   339,   758,  1491,  ...
  10,  96,  462,  1558,  4194,  9660,  ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.

Main diagonal gives A292631.

Programs

Maple

[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

Mathematica

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

Formula

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