A306335 Expansion of e.g.f. BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)).

1, 1, 1, -1, 4, -21, 133, -981, 8244, -77694, 811194, -9292075, 115843000, -1561272571, 22618147199, -350481556959, 5784147674772, -101284047800632, 1875504207906184, -36616289396963678, 751702523788615816, -16187581390548113842, 364861626149143519378, -8590429045711448354359

Offset: 0

Author

Ilya Gutkovskiy, Feb 08 2019

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A001405, A048994, A086672, A305560.

Programs

Maple

E:= BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)):
S:= series(E,x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Feb 10 2019

Mathematica

nmax = 23; CoefficientList[Series[BesselI[0, 2 Log[1 + x]] + BesselI[1, 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 23}]

PARI

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(k, k\2)); \\ Michel Marcus, Feb 09 2019

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001405(k).