A336804 a(n) = (n!)^2 * Sum_{k=0..n} 2^(n-k) / (k!)^2.

1, 3, 25, 451, 14433, 721651, 51958873, 5091969555, 651772103041, 105587080692643, 21117416138528601, 5110414705523921443, 1471799435190889375585, 497468209094520608947731, 195007537965052078707510553, 87753392084273435418379748851, 44929736747147998934210431411713

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2021

Keywords

• nonn

Crossrefs

Cf. A006040, A010844, A228513, A336805, A336807, A336808.

Programs

Mathematica

Table[n!^2 Sum[2^(n - k)/k!^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 2*x).

a(0) = 1; a(n) = 2 * n^2 * a(n-1) + 1.