A061573 -id:A061573 - cp's OEIS frontend

A061572 a(n) = (n!)^2 * Sum_{k=1..n} 1/(k^2*(k-1)!).

1, 5, 47, 758, 18974, 683184, 33476736, 2142516144, 173543847984, 17354385161280, 2099880608143680, 302382807612606720, 51102694487009537280, 10016128119460096327680, 2253628826878608852019200, 576928979680925173791283200, 166732475127787396148470732800

Offset: 1

N. J. A. Sloane, May 19 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A061573.

a(n) = { n!^2*sum(k=1, n, 1/(k^2*(k-1)!)) } \\ Harry J. Smith, Jul 24 2009

Recurrence: a(1) = 1, a(2) = 5, a(n) = (n^2+n-1)*a(n-1) - (n-1)^3*a(n-2) for n >= 3. The sequence b(n) = n!^2 also satisfies this recurrence with the initial conditions b(1) = 1 and b(2) = 4. Hence we have the finite continued fraction expansion a(n)/b(n) = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)))). Lim n -> infinity a(n)/b(n) = Ei(1) - gamma = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)-...))). Cf. A061573. - Peter Bala, Jul 10 2008

A346405 a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k)^2 * k!).

Original entry on oeis.org

0, 1, 5, 31, 268, 3476, 70656, 2202432, 98622336, 5954736384, 463100042880, 44924476970880, 5308404719823360, 749930460864929280, 124754522068412651520, 24129984694192721971200, 5368254991077002482483200, 1360938718277588430567014400, 389980903967231535140578099200

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Cf. A001819, A002104, A002720, A061573, A346409.

Table[(n!)^2 Sum[1/((n - k)^2 k!), {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * exp(x).

cp's OEIS Frontend

A061572 a(n) = (n!)^2 * Sum_{k=1..n} 1/(k^2*(k-1)!).

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