A340888 -id:A340888 - 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))).

A340887 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 3^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 7, 99, 2511, 99531, 5680125, 441226521, 44766049599, 5748319130283, 911271895816077, 174799606363478361, 39903413238125862309, 10690643656077551475921, 3321750648705212259711063, 1184831658624977151885176859, 480843465699932167142334581919

Offset: 0

Ilya Gutkovskiy, Jan 25 2021

nonn

Cf. A102221, A255927, A340886, A340888.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 3^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[3/(4 - BesselI[0, 2 Sqrt[3 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 3 / (4 - BesselI(0,2*sqrt(3*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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