A351811 -id:A351811

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page 1

A351810

G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 4*x)) / (1 - 4*x)^2.

+10
2

1, 1, 9, 69, 565, 5305, 56929, 680685, 8902349, 126121313, 1923133433, 31379181461, 544931376229, 10024917092105, 194602995875985, 3972686705253181, 85035210652191485, 1903471938128641457, 44453001710603619369, 1080789854059236415973, 27304602412815047204501

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

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

MATHEMATICA

nmax = 20; A[_] = 0; Do[A[x_] = 1 + x A[x/(1 - 4 x)]/(1 - 4 x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

CROSSREFS

Cf. A004213, A040027, A326324, A351756, A351757, A351811, A351812.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 19 2022

STATUS

approved

A351812

G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 6*x)) / (1 - 6*x)^2.

+10
2

1, 1, 13, 139, 1531, 19021, 271453, 4358179, 76896931, 1471496341, 30333401893, 670125430219, 15784342627531, 394467249489661, 10415430504486733, 289527454704656659, 8447556960083354131, 258008113711846390981, 8228947382557338981973, 273472796359924298018299

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 6^(k-1) * a(n-k).

MATHEMATICA

nmax = 19; A[_] = 0; Do[A[x_] = 1 + x A[x/(1 - 6 x)]/(1 - 6 x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

CROSSREFS

Cf. A005012, A040027, A351756, A351757, A351810, A351811.