A322240 -id:A322240 - cp's OEIS frontend

A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008
Ignoring initial term, equals the logarithmic derivative of A091147. - Paul D. Hanna, Dec 08 2018
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..1433 (terms 0..200 from Vincenzo Librandi)
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.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Cf. A002426, A084600-A084604, A084606-A084615.

Cf. A322240 (a(n)^2), A091147.

a := n -> simplify(2^n*GegenbauerC(n,-n, -1/4)):
seq(a(n), n=0..22); # Peter Luschny, May 08 2016

Table[n!*SeriesCoefficient[E^x*BesselI[0,4*x],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 16];
Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 18 2018 *)

for(n=0,30,t=polcoeff((1+x+4*x^2)^n,n,x); print1(t","))
for(n=0,20,print1(a(n),", "))

{a(n) = sum(k=0,n, (-3)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Dec 09 2018

E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) is also the central coefficient of (4+x+x^2)^n; a(n) = Sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907. - N-E. Fahssi, Mar 30 2008
a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). - Peter Luschny, Sep 12 2011
D-finite with recurrence a(n+2) = ((2*n+3)*a(n+1) + 15*(n+1)*a(n))/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012
a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 2^n*GegenbauerC(n, -n, -1/4). - Peter Luschny, May 08 2016
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 16). - Peter Luschny, Mar 18 2018
a(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k). - Paul D. Hanna, Dec 09 2018
a(n) = Sum_{k=0..n} 5^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k). - Seiichi Manyama, May 01 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A307810 Expansion of 1/AGM(1-64x, sqrt((1-16x)(1-256x))).

Original entry on oeis.org

1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400

Offset: 0

Seiichi Manyama, Apr 30 2019

nonn

See A246923.

Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..416

Cf. A307695.

(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))

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

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

a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.

a(n) ~ 2^(8*n+2) / (3*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A307811 Expansion of 1/AGM(1-45x, sqrt((1-25x)(1-81x))).

Original entry on oeis.org

1, 49, 2601, 148225, 8970025, 570111129, 37678303881, 2567836387809, 179267329355625, 12754414737348025, 921185098227422161, 67340346156989933769, 4971327735657992896201, 369994703739586257235225, 27725052308247030792515625, 2089567204521186409129541025

Offset: 0

Seiichi Manyama, Apr 30 2019

nonn

See A246923.

Also the squares of coefficients in g.f. 1/sqrt((1-5*x)*(1-9*x)).

Seiichi Manyama, Table of n, a(n) for n = 0..525

Cf. A104454.

(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), this sequence (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), A307810 (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).

a[n_] := Sum[5^(n-k) * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 16, 0] // Flatten (* Amiram Eldar, May 13 2021 *)

N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))

{a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}

{a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}

a(n) = A104454(n)^2 = (Sum_{k=0..n} 5^(n-k)*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 9^(n-k)*(-1)^k*binomial(n,k)*binomial(2k,k))^2.

a(n) ~ 3^(4*n+2) / (4*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

Original entry on oeis.org

1, 1, 41, 249, 6305, 77569, 1665321, 27724889, 574252417, 10958980929, 228679916905, 4671350051321, 99292476904609, 2107949882690241, 45658568907254505, 993562984208479193, 21876513296218002433, 484448162130512673665, 10812975015547281792937, 242647271141110287979513, 5477046865641884201456033

Offset: 0

Paul D. Hanna, Dec 08 2018

nonn

Compare to: exp( Sum_{n>=1} A084605(n) * x^n/n ) = (1-x - sqrt(1 - 2*x - 15*x^2))/(8*x^2), the g.f. of A091147.

Sequence A322240(n) = A084605(n)^2 has generating function 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ).

			G.f.: A(x) = 1 + x + 41*x^2 + 249*x^3 + 6305*x^4 + 77569*x^5 + 1665321*x^6 + 27724889*x^7 + 574252417*x^8 + 10958980929*x^9 + 228679916905*x^10 + ...
such that
log(A(x)) = x + 81*x^2/2 + 625*x^3/3 + 21025*x^4/4 + 314721*x^5/5 + 8071281*x^6/6 + 155975121*x^7/7 + 3685097025*x^8/8 + ... + A084605(n)^2 * x^n/n + ...
RELATED SERIES.
The g.f. of A084605 equals the series
1/sqrt(1 - 2*x - 15*x^2) = 1 + x + 9*x^2 + 25*x^3 + 145*x^4 + 561*x^5 + 2841*x^6 + 12489*x^7 + 60705*x^8 + 281185*x^9 + ... + A084605(n) * x^n/n + ...

Cf. A322240, A084605.

{a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^m)), m)^2 *x^m/m)+x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))

cp's OEIS Frontend

A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

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A307810 Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).

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A307811 Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).

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A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

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A307810 Expansion of 1/AGM(1-64x, sqrt((1-16x)(1-256x))).

A307811 Expansion of 1/AGM(1-45x, sqrt((1-25x)(1-81x))).