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A336808

a(n) = (n!)^2 * Sum_{k=0..n} 5^(n-k) / (k!)^2.

1, 6, 121, 5446, 435681, 54460126, 9802822681, 2401691556846, 768541298190721, 311259225767242006, 155629612883621003001, 94155915794590706815606, 67792259372105308907236321, 57284459169428986026614691246, 56138769986040406306082397421081, 63156116234295457094342697098716126

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..15.

FORMULA

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 5*x).

a(0) = 1; a(n) = 5 * n^2 * a(n-1) + 1.

a(n) ~ n!^2 * BesselI(0, 2/sqrt(5)) * 5^n. - Vaclav Kotesovec, Feb 09 2026

MATHEMATICA

Table[n!^2 Sum[5^(n - k)/k!^2, {k, 0, n}], {n, 0, 15}]

nmax = 15; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2