A336210 -id:A336210 - cp's OEIS frontend

A336209 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

1, -1, 1, 2, -15, -46, 880, 5837, -132783, -2109238, 35966256, 1440196097, -8909037720, -1504006716551, -16097564749643, 2021100230840147, 83130656159529937, -2475528081920694566, -331363460045748820376, -3874344448291316066455, 1255007424437046915956520

Offset: 0

Ilya Gutkovskiy, Jul 12 2020

sign

Cf. A000587, A023998, A336210.

a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[-Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^2).

a(n) = (n!)^2 * [x^n] exp(1 - BesselI(0,2*sqrt(x))).

A336831 a(n) = (n!)^n * [x^n] exp(-Sum_{k>=1} (-x)^k / (k!)^n).

Original entry on oeis.org

1, 1, 1, 10, 4359, 91406876, 111657668637280, 11436881770074767723291, 137560155520600195617494951186559, 260122627893213770028102613184254361777327032, 99781796293430843492956500115058179262448159117567276656136

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Cf. A000587, A275044, A336209, A336210, A336439.

Table[(n!)^n SeriesCoefficient[Exp[-Sum[(-x)^k/(k!)^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := b[n, k] = If[n == 0, 1, -Sum[(-1)^(n - j) Binomial[n, j]^k (n - j) b[j, k], {j, 0, n - 1}]/n]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

cp's OEIS Frontend

A336209 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

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