A351046 a(1)=1; a(2)=4; for n>2, a(n) = a(n-1) + A000217(n)*a(n-2).

1, 4, 10, 50, 200, 1250, 6850, 51850, 360100, 3211850, 26978450, 277502750, 2732541700, 31870330450, 359775334450, 4694140275650, 59739766446500, 862437753582650, 12212993378417650, 193324921630774150, 3014526392045251300, 51925731564631111250, 883935015769120470050

Offset

1,2

Links

Table of n, a(n) for n=1..23.

Formula

E.g.f. A(x) satisfies the differential equation 6*A(x) + (6*x + 2)*A'(x) + (x^2 - 2)*A''(x) = 0, A(0) = 1, A'(0) = 1.

a(n) ~ n! * c * n^(1 + 1/sqrt(2)) / 2^(n/2), where c = 0.42906449224324091038170340685604072807700713285504473...

Mathematica

RecurrenceTable[{a[1]==1, a[2]==4, a[n]==a[n-1] + n*(n+1)/2*a[n-2]}, a, {n, 1, 20}]
nmax = 20; Round[Rest[CoefficientList[Series[(Sqrt[2]*(2 + x^2) - 4*x) * (((2 - x^2)^2 * (1 - 3*Sqrt[2]) * Hypergeometric2F1[2, 3, 3 - 1/Sqrt[2], (2 + Sqrt[2]*x)/4] + 16*(1 + x) * ((Sqrt[2] + x)/(Sqrt[2] - x))^(1/ Sqrt[2]) * ((1 - 3*Sqrt[2]) * Hypergeometric2F1[2, -1/Sqrt[2], 3 - 1/Sqrt[2], -1] + 6*Hypergeometric2F1[3, -1/Sqrt[2], 4 - 1/Sqrt[2], -1])) / (8*(Sqrt[2] - x)^2 * (2 - x^2)^2 * (3*Sqrt[2] * Hypergeometric2F1[3, -1/Sqrt[2], 4 - 1/Sqrt[2], -1] - (6 - Sqrt[2]) * Hypergeometric2F1[2, -1/Sqrt[2], 3 - 1/Sqrt[2], -1]))), {x, 0, nmax}], x]] * Range[nmax]!]

Crossrefs

Cf. A166474.

Sequence in context: A173086 A081565 A151611 * A208236 A032495 A109387

Adjacent sequences: A351043 A351044 A351045 * A351047 A351048 A351049

Keyword

nonn

Author

Vaclav Kotesovec, Jan 30 2022, following a suggestion from John M. Campbell

Status

approved