A317310 Expansion of e.g.f. (1 + x)^2*BesselI(0,2*log(1 + x)).

1, 2, 4, 6, 4, 0, -2, 14, -100, 792, -6996, 68508, -737882, 8676200, -110627142, 1520662410, -22418697948, 352885526856, -5907074659016, 104782694989616, -1963418893492364, 38753471698684512, -803656781974363412, 17469671114170029708, -397223288562294817330, 9429329994809282773300

Offset: 0

Author

Ilya Gutkovskiy, Jan 22 2019

Keywords

• sign

Crossrefs

Cf. A000984, A048994, A086672, A305406.

Programs

Maple

a:=series((1 + x)^2*BesselI(0,2*log(1 + x)), x=0, 26): seq(n!*coeff(a, x, n), n=0..25); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 25; CoefficientList[Series[(1 + x)^2 BesselI[0, 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Binomial[2 k, k], {k, 0, n}], {n, 0, 25}]

PARI

my(x='x + O('x^30)); Vec(serlaplace((1 + x)^2*besseli(0,2*log(1 + x)))) \\ Michel Marcus, Mar 27 2019

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000984(k).