A346291 -id:A346291 - cp's OEIS frontend

A346292 a(0) = 1; a(n) = (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

1, 0, 0, 4, 36, 576, 17600, 694800, 35802144, 2391438336, 200018045952, 20476348214400, 2521840589347200, 368057828019898368, 62841061478699292672, 12413136137144581203456, 2809529229255558769612800, 722458985698006017844838400, 209487621780682072569567903744

Offset: 0

Ilya Gutkovskiy, Jul 13 2021

nonn

Cf. A038205, A074707, A346291.

a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 3, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x - x^2 / 4 ).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=3} x^n / n^2 ).

A346372 a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

Original entry on oeis.org

1, 1, 2, 10, 124, 2396, 64856, 2452472, 124483360, 8146185504, 668645524032, 67374446014272, 8183368905811584, 1179807474740449920, 199266648878034317568, 38984601149045449948416, 8748103140554862876727296, 2232274640259371687436982272, 642805438643602793466093711360

Offset: 0

Ilya Gutkovskiy, Jul 14 2021

nonn

Cf. A000266, A074707, A346291.

a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[(Binomial[n, k] k!)^2 a[n - k]/k, {k, 3, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[PolyLog[2, x] - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( polylog(2,x) - x^2 / 4 ).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / n^2 ).

cp's OEIS Frontend

A346292 a(0) = 1; a(n) = (1/n) * Sum_{k=3..n} (binomial(n,k) * k!)^2 * a(n-k) / k.

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