A322246 -id:A322246 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, Dec 10 2018

nonn

			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 + ...

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

Cf. A246467, A246906, A246923, A322246.

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 *)

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

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

A098658 a(n) = 3^n(2n)!/(n!)^2.

Original entry on oeis.org

1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070, 956987460, 10909657044, 124965162504, 1437099368796, 16581915793800, 191876454185400, 2225766868550640, 25874539846901190, 301362287628613860

Offset: 0

Paul Barry, Sep 20 2004

Number of lattice paths from (0,0) to (n,n) using steps (0,1) and three kinds of steps (1,0). - Joerg Arndt, Jul 01 2011
Sixth binomial transform of 1/sqrt(1-36*x^2).
Diagonal of the rational function 1 / (1 - 3*x - y). - Ilya Gutkovskiy, Apr 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A000984, A059304, A101600, A322246.

[3^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

Table[3^n (2n)!/(n!)^2,{n,0,20}] (* Harvey P. Dale, Dec 14 2011 *)

/* same as in A092566 but use */
steps=[[1,0], [1,0], [1,0], [0,1]]; /* note the triple [1,0] */
/* Joerg Arndt, Jun 30 2011 */

G.f.: 1/sqrt((1-6*x)^2-36*x^2) = 1/sqrt(1-12*x).
E.g.f.: exp(6*x)*BesselI(0, 6x).
a(n) = [t^n](1+6*t+9*t^2)^n.
a(n) = 3^n*A000984(n). - R. J. Mathar, Oct 10 2012
G.f.: Q(0), where Q(k) = 1 + 12*x*(4*k+1)/( 4*k+2 - 12*x*(4*k+2)*(4*k+3)/(12*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
n*a(n) +6*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 27 2014
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 45*x^2 + 378*x^3 + ... is the o.g.f. for A101600. - Peter Bala, Jul 16 2015
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 12/11 + 12*sqrt(11)*arcsin(1/sqrt(12))/121.
Sum_{n>=0} (-1)^n/a(n) = 12/13 - 12*sqrt(13)*arcsinh(1/sqrt(12))/169. (End)
From Peter Bala, Oct 12 2024: (Start)
a(n) = Integral_{x = 0..12} x^n * w(x) dx, where w(x) = 1/( Pi*sqrt(x*(12 - x)) ) is positive on the interval (0, 12). The weight function w(x) is singular at x = 0 and at x = 12 and is the solution of the Hausdorff moment problem.
Binomial transform of A322246.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r. (End)

A340973 Generating function Sum_{n >= 0} a(n)x^n = 1/sqrt((1-x)(1-13*x)).

Original entry on oeis.org

1, 7, 67, 721, 8179, 95557, 1137709, 13725439, 167204947, 2052215893, 25338173497, 314356676179, 3915672171229, 48938691421627, 613404577267843, 7707619156442401, 97058716523798227, 1224551690144551237

Offset: 0

Seiichi Manyama, Feb 01 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..899

Column k=3 of A340970.

Cf. A322246, A337167.

a[n_] := Sum[3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Feb 01 2021 *)
nxt[{n_,a_,b_}]:={n+1,b,(7*b(2n+1)-13*n*a)/(n+1)}; Join[{1},NestList[nxt,{2,7,67},20] [[All,2]]] (* Harvey P. Dale, Apr 27 2022 *)

my(N=20, x='x+O('x^N)); Vec(1/sqrt((1-x)*(1-13*x)))

a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*k, k));

PARI
```
a(n) = polcoef((1+7*x+9*x^2)^n, n);
```

a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(2*k,k).
a(n) = [x^n] (1+7*x+9*x^2)^n.
n * a(n) = 7 * (2*n-1) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1.
E.g.f.: exp(7*x) * BesselI(0,6*x). - Ilya Gutkovskiy, Feb 01 2021
a(n) ~ 13^(n + 1/2) / (2 * sqrt(3*Pi*n)). - Vaclav Kotesovec, Nov 13 2021
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(n,k) * binomial(2*k,k). (End)

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200

Offset: 0

Seiichi Manyama, Apr 22 2019

nonn

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..831

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))

{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}

{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019

A340972 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k) * binomial(2*k,k).

Original entry on oeis.org

1, -1, 17, -395, 13345, -592299, 32630401, -2148740061, 164682639745, -14401797806195, 1415344434226801, -154426458074411313, 18523291145011712929, -2422743610992855309925, 343167234011405980982625

Offset: 0

Seiichi Manyama, Feb 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..322

Cf. A322246, A339001, A340971.

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)

a(n) = sum(k=0, n, (-n)^k*binomial(n, k)*binomial(2*k, k));

a(n) = polcoef(1/sqrt((1-x)*(1+(4*n-1)*x)+x*O(x^n)), n);

a(n) = polcoef((1-(2*n-1)*x+(n*x)^2)^n, n);

a(n) = [x^n] 1/sqrt((1-x)*(1+(4*n-1)*x)).
a(n) = [x^n] (1-(2*n-1)*x+(n*x)^2)^n.
a(n) = n! * [x^n] BesselI(0,2*n*x) / exp((2*n-1)*x). - Ilya Gutkovskiy, Feb 01 2021
a(n) ~ (-1)^n * exp(-1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 13 2021
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} (-4*n+1)^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} n^k * (-4*n+1)^(n-k) * binomial(n,k) * binomial(2*k,k). (End)

cp's OEIS Frontend

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

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A098658 a(n) = 3^n*(2*n)!/(n!)^2.

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A340973 Generating function Sum_{n >= 0} a(n)*x^n = 1/sqrt((1-x)*(1-13*x)).

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A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

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A340972 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k) * binomial(2*k,k).

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

A098658 a(n) = 3^n(2n)!/(n!)^2.

A340973 Generating function Sum_{n >= 0} a(n)x^n = 1/sqrt((1-x)(1-13*x)).

A307695 Expansion of 1/(sqrt(1-4x)sqrt(1-16*x)).