A372018 -id:A372018

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A372019

G.f. A(x) satisfies A(x) = ( 1 + 9*x*A(x)/(1 - x*A(x)) )^(1/3).

+10
3

1, 3, 3, 3, 30, 57, 84, 867, 1893, 3162, 33132, 76953, 136812, 1446204, 3478764, 6420387, 68260134, 167946159, 317782524, 3392340186, 8479140510, 16332164868, 174873206424, 442212416121, 863222622780, 9264327739716, 23637757714788, 46624054987452

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..27.

FORMULA

a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+1/3,k) * binomial(n-1,n-k).

D-finite with recurrence n*(n-1)*(n+1)*a(n) -8*(2*n-5)*(8*n^2-40*n+57)*a(n-3) +4096*(n-5)*(n-6)*(n-4)*a(n-6)=0. - R. J. Mathar, Apr 22 2024

MAPLE

A371019 := proc(n)

add(9^k*binomial((n+1)/3, k)*binomial(n-1, k-1), k=0..n) ;

%/(n+1) ;

end proc:

seq(A371019(n), n=0..60) ; # R. J. Mathar, Apr 22 2024

PROG

(PARI) a(n) = sum(k=0, n, 9^k*binomial(n/3+1/3, k)*binomial(n-1, n-k))/(n+1);

CROSSREFS

Cf. A372018, A372020.

Cf. A372004.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 15 2024

STATUS

approved

A372020

G.f. A(x) satisfies A(x) = ( 1 + 16*x*A(x)/(1 - x*A(x)) )^(1/4).

+10
2

1, 4, -4, 4, 156, -1212, 5628, 196, -251620, 2500484, -12608772, 16004, 671151260, -7039845180, 37258827516, 1585476, -2133978944740, 23052545651460, -125166709730820, 174117124, 7480512144282780, -82265332158299580, 453899597102224380, 20390254020

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..23.

FORMULA

a(n) = (1/(n+1)) * Sum_{k=0..n} 16^k * binomial(n/4+1/4,k) * binomial(n-1,n-k).

PROG

(PARI) a(n) = sum(k=0, n, 16^k*binomial(n/4+1/4, k)*binomial(n-1, n-k))/(n+1);

CROSSREFS

Cf. A372018, A372019.

Cf. A372005.