A352467 -id:A352467 - cp's OEIS frontend

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2n,2k)^3 * a(n-k).

1, 1, 217, 735751, 16225658905, 1485378967457251, 429009059656530602767, 324779065084721999818137709, 563805297587600177760431368896025, 2028620600892240327820781003315525267467, 13978450121866685445815888094629703793828769467

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A094088, A336195, A352467, A352471.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k]^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 10}]
nmax = 20; Take[CoefficientList[Series[1/(1 - Sum[x^(2 k)/(2 k)!^3, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3, {1,-1, 2}]

Sum_{n>=0} a(n) * x^(2*n) / (2*n)!^3 = 1 / (1 - Sum_{n>=1} x^(2*n) / (2*n)!^3).

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2k+1)^2 a(n-2*k-1).

Original entry on oeis.org

1, 1, 4, 37, 608, 15601, 576472, 28993693, 1904637184, 158352856129, 16253786050904, 2018684970206653, 298373110433984192, 51757706826973479697, 10412613242348421164400, 2404755328388872932588037, 631887117002962512609921024, 187441600433239155105076467457

Offset: 0

Ilya Gutkovskiy, Mar 17 2022

nonn

Cf. A006154, A102221, A346220, A352467, A352471.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^2 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^2, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - Sum_{n>=0} x^(2*n+1) / (2*n+1)!^2).

Sum_{n>=0} a(n) * x^n / n!^2 = 1 / (1 - (BesselI(0,2*sqrt(x)) - BesselJ(0,2*sqrt(x))) / 2).

cp's OEIS Frontend

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2n,2k)^3 * a(n-k).

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Formula

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2k+1)^2 a(n-2*k-1).

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cp's OEIS Frontend

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k)^3 * a(n-k).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * a(n-2*k-1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A352468 a(0) = 1; a(n) = Sum_{k=1..n} binomial(2n,2k)^3 * a(n-k).

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2k+1)^2 a(n-2*k-1).