A370474 G.f. A(x) satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(3/2)).

1, 2, 9, 54, 372, 2778, 21873, 178786, 1502649, 12904524, 112741664, 998871030, 8953443276, 81047485148, 739846170864, 6803054508702, 62954736555836, 585850907166084, 5479077065774682, 51470699845616004, 485456696541512442, 4595280949098247422

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..993

Formula

a(n) = Sum{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1,n)/(3*n/2+3*k/2+1).

From Seiichi Manyama, Dec 12 2024: (Start)

G.f. A(x) satisfies:

(1) A(x) = ( 1 + x*A(x)^(5/2)/(1 + x*A(x)^(3/2)) )^2.

(2) A(x) = 1/( 1 - x*A(x)^2/(1 + x*A(x)^(3/2)) )^2.

(3) A(x) = B(x)^2 where B(x) is the g.f. of A271469.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

D-finite with recurrence: (1458*n^4 + 4374*n^3 + 4212*n^2 + 1296*n)*a(n) + (-54675*n^4 - 382968*n^3 - 998325*n^2 - 1127412*n - 457380)*a(n + 1) + (282636*n^4 + 1874988*n^3 + 3224772*n^2 - 1159740*n - 4745160)*a(n + 2) + (889326*n^4 + 18310140*n^3 + 122844918*n^2 + 339978192*n + 336576168)*a(n + 3) + (-8487504*n^4 - 145041744*n^3 - 929861088*n^2 - 2650917816*n - 2835872208)*a(n + 4) + (8322304*n^4 + 172017664*n^3 + 1332072824*n^2 + 4579893704*n + 5898234240)*a(n + 5) + (-112128*n^4 - 264960*n^3 + 20353440*n^2 + 175865520*n + 410584608)*a(n + 6) + (-61440*n^4 - 1689600*n^3 - 17399040*n^2 - 79516800*n - 136080000)*a(n + 7) = 0. - Robert Israel, Feb 25 2026

Maple

f:= gfun:-rectoproc({(1458*n^4 + 4374*n^3 + 4212*n^2 + 1296*n)*a(n) + (-54675*n^4 - 382968*n^3 - 998325*n^2 - 1127412*n - 457380)*a(n + 1) + (282636*n^4 + 1874988*n^3 + 3224772*n^2 - 1159740*n - 4745160)*a(n + 2) + (889326*n^4 + 18310140*n^3 + 122844918*n^2 + 339978192*n + 336576168)*a(n + 3) + (-8487504*n^4 - 145041744*n^3 - 929861088*n^2 - 2650917816*n - 2835872208)*a(n + 4) + (8322304*n^4 + 172017664*n^3 + 1332072824*n^2 + 4579893704*n + 5898234240)*a(n + 5) + (-112128*n^4 - 264960*n^3 + 20353440*n^2 + 175865520*n + 410584608)*a(n + 6) + (-61440*n^4 - 1689600*n^3 - 17399040*n^2 - 79516800*n - 136080000)*a(n + 7), a(0) = 1, a(1) = 2, a(2) = 9, a(3) = 54, a(4) = 372, a(5) = 2778, a(6) = 21873}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 25 2026

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n/2+3*k/2+1, n)/(3*n/2+3*k/2+1));
(PARI) a(n, r=2, s=-1, t=5, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024

Crossrefs

Cf. A006013, A370475.

Cf. A262441, A366400.

Cf. A271469.

Sequence in context: A223943 A371698 A241125 * A089436 A368178 A394163

Adjacent sequences: A370471 A370472 A370473 * A370475 A370476 A370477

Keyword

nonn

Author

Seiichi Manyama, Mar 31 2024

Status

approved