A322247 a(n) = A322246(n)^2, the square of the central coefficient in (1 + 5*x + 9*x^2)^n.

1, 25, 1849, 156025, 14523721, 1426950625, 145317252025, 15178231605625, 1615509001626025, 174471431239950625, 19062335608125901729, 2102483602307417980225, 233721380163477368733481, 26154175972512598202392225, 2943361280244176889333396889, 332869229155486455718147125625, 37806108834415039621850996946025, 4310099976506176089944803738530625, 493021434686696395739629566004713025

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..481

Formula

G.f.: 1 / AGM(1 + 11*x, sqrt((1 - x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.

G.f.: 1 / AGM((1-x)*(1-11*x), (1+x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * binomial(n,k)*binomial(2*k,k).

a(n) ~ 11^(2*n + 1) / (12*Pi*n). - Vaclav Kotesovec, Dec 13 2018

Example

G.f.: A(x) = 1 + 25*x + 1849*x^2 + 156025*x^3 + 14523721*x^4 + 1426950625*x^5 + 145317252025*x^6 + 15178231605625*x^7 + 1615509001626025*x^8 + ...
that is,
A(x) = 1 + 5^2*x + 43^2*x^2 + 395^2*x^3 + 3811^2*x^4 + 37775^2*x^5 + 381205^2*x^6 + 3895925^2*x^7 + 40193395^2*x^8 + 417697775^2*x^9 + ... + A322246(n)^2*x^n + ...

Mathematica

a[n_] := Sum[(-1)^(n-k) * 3^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)

Prog

(PARI) /* a(n) = A322246(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt( (1 - x)*(1 + 11*x) +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322246(n)^2 - g.f. */
{a(n) = polcoeff( (1 + 5*x + 9*x^2 +x*O(x^n))^n, n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322246(n)^2 - binomial sum */
{a(n) = sum(k=0, n, (-1)^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* a(n) = A322246(n)^2 - binomial sum */
{a(n) = sum(k=0, n, 11^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 1*11*x, sqrt((1 - 1^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A246467, A246906, A246923, A322246.

Sequence in context: A370436 A172261 A023113 * A177837 A056047 A281436

Adjacent sequences: A322244 A322245 A322246 * A322248 A322249 A322250

Keyword

nonn

Author

Paul D. Hanna, Dec 10 2018

Status

approved