A340887 -id:A340887 - cp's OEIS frontend

A340886 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 2^(n-k-1) * a(k).

1, 1, 6, 76, 1720, 60816, 3096384, 214579296, 19422473088, 2224980891904, 314675568756736, 53849929134122496, 10966912240761425920, 2621246193301011159040, 726608751113679704248320, 231217063994112487051984896, 83713709650818121936828858368

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A122704, A340887, A340888.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 2^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[2/(3 - BesselI[0, 2 Sqrt[2 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 2 / (3 - BesselI(0,2*sqrt(2*x))).

A340888 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 4^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 8, 124, 3456, 150656, 9453056, 807373568, 90066059264, 12716049596416, 2216452086693888, 467465806422867968, 117332539562036035584, 34562989958399757647872, 11807922834511544081973248, 4630865359842075866336067584, 2066370767828213666946077425664

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A326324, A340886, A340887.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 4^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[4/(5 - BesselI[0, 4 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 4 / (5 - BesselI(0,4*sqrt(x))).

cp's OEIS Frontend

A340886 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 2^(n-k-1) * a(k).

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