A383716 -id:A383716 - cp's OEIS frontend

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

1, 6, 46, 378, 3214, 27896, 245506, 2182396, 19548046, 176142312, 1594831736, 14497410186, 132224930146, 1209397179048, 11088872706188, 101890087382168, 937973964234638, 8649109175873288, 79872298511230120, 738583466508887304, 6837944227813170424

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000302, A160906, A386812.
Cf. A005810, A078995, A147855, A226733, A226761, A385605.
Cf. A098430, A383716.
Cf. A066381.

[&+[Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025

Table[Sum[Binomial[4*n+1,k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

a(n) = sum(k=0, n, binomial(4*n+1, k));

a(n) = [x^n] 1/((1-2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+k,k).
D-finite with recurrence +645*n*(3*n-1)*(3*n-2)*a(n) +8*(-56722*n^3+213090*n^2-305978*n+150255)*a(n-1) +128*(62908*n^3-282348*n^2+385070*n-126735)*a(n-2) +12288*(-2486*n^3+8918*n^2+758*n-18935)*a(n-3) -2949120*(2*n-7)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(4*n+1) - binomial(4*n+1, n)*(hypergeom([1, -1-3*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k). - Seiichi Manyama, Aug 07 2025
a(n) ~ 2^(8*n + 3/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Aug 07 2025
G.f.: g^2/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^2*(8-2*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 16 2025

A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

Original entry on oeis.org

1, 5, 31, 209, 1471, 10625, 78079, 580865, 4361215, 32978945, 250806271, 1916280833, 14698053631, 113104519169, 872801042431, 6751535300609, 52337071357951, 406468580343809, 3162019821780991, 24634626678980609, 192179216026959871

Offset: 0

Joseph Abate, Jun 15 2010

Hankel transform is A133460.

			a(3) = (1,4,10,20)dot(1,2,4,8) = 209.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian motion and generalized Catalan numbers, Journal of Integer Sequences, Vol. 14 (2011).
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.

Row sums of A091811.

Cf. A383326, A383716.

a := n -> binomial(2*n+2,n+1)*hypergeom([-n, n + 1], [n + 2], -1)/2:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 21 2017

CoefficientList[Series[(3-Sqrt[1-8*x])/(2*(1+x)*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
Table[Sum[2^k*Binomial[n + k, k], {k, 0, n}], {n, 0, 20}] (* Michael De Vlieger, Oct 28 2016 *)
a[n_] := (-1)^(n + 1) - 2^(n + 1) (2n + 1) Binomial[2n, n] Hypergeometric2F1[1, 2n + 2, n + 2, 2]/(n + 1); Array[a, 22, 0] (* Robert G. Wilson v, Jul 21 2018 *)

a(n) = Sum_{k = 0..n} A046899(n,k)*2^k = Sum_{k = 0..n} 2^k * binomial(n+k,k).
G.f.: (1/3)*(4/sqrt(1 - 8*x) - 1/(1 - x*c(2*x))) with c(x) the g.f. of the Catalan numbers A000108.
a(n) = (1/3)*(4*2^n*A000984(n) - A064062(n)).
a(n) + a(n+1) = 6*2^n*A001700(n).
O.g.f.: (3 - sqrt(1 - 8*x))/(2*(1 + x)*sqrt(1 - 8*x)). - Peter Bala, Apr 10 2012
a(n) = 2^n *binomial(2+2*n,1+n)*2F1(1, 2+2*n; 2+n;-1). - Olivier Gérard, Aug 19 2012
D-finite with recurrence n*a(n) = (7*n - 4)*a(n-1) + 4*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 2^(3n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = Sum_{k = 0..n} binomial(k+n,n) * binomial(2*n+1,n-k). - Vladimir Kruchinin, Oct 28 2016
a(n) = 1/2*(n + 1)*binomial(2*n+2,n+1)*Sum_{k = 0..n} binomial(n,k)/(n + k + 1). - Peter Bala, Feb 21 2017
a(n) = binomial(2*n+2,n+1)*hypergeom([-n, n+1], [n+2], -1)/2. - Peter Luschny, Feb 21 2017
a(n) = (-1)^(n+1) - 2^(n+1)*(2*n+1)*binomial(2*n,n)*hypergeom([1, 2*n+2], [n+2], 2)/(n+1). - John M. Campbell, Jul 14 2018
From Akiva Weinberger, Dec 06 2024: (Start)
a(n) = (2*n + 1)!/(n!^2) * Integral_{t=0..1} (t + t^2)^n dt.
a(n) = (Integral_{t=0..1} (t + t^2)^n dt) / (Integral_{t=0..1} (t - t^2)^n dt). (End)
a(n) = (-1)^n * Sum_{k=0..n} binomial(2*n+1,k) * (-2)^k. - André M. Timpanaro, Dec 15 2024
From Seiichi Manyama, Aug 04 2025: (Start)
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(n+1)). (End)

A383326 a(n) = Sum_{k=0..n} binomial(3n+1,k) binomial(3*n-k,n-k).

Original entry on oeis.org

1, 7, 71, 799, 9439, 114687, 1419263, 17791487, 225172991, 2870945791, 36819740671, 474470776831, 6138443497471, 79681448443903, 1037278449106943, 13536444411412479, 177030837540093951, 2319618918724403199, 30444928900076666879, 400189735705069486079, 5267487129636270243839

Offset: 0

Seiichi Manyama, Aug 04 2025

nonn

Cf. A178792, A383716.

Cf. A370097.

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

a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k).

A386834 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 8, 111, 1738, 28701, 488412, 8473387, 148994510, 2645999673, 47349481408, 852429930567, 15421507805106, 280126256513109, 5105764838932388, 93331970924544099, 1710369544783134614, 31412304686874624113, 578023658034894471048, 10654486069487503147135

Offset: 0

Seiichi Manyama, Aug 05 2025

nonn

Cf. A116881, A386833.
Cf. A370101, A386837.
Cf. A383716.

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

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(3*n)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(3*n+k-1,k).
Conjecture D-finite with recurrence +3*n*(16543753*n -26995933)*(3*n-1)*(3*n-2)*a(n) +(-1669899251*n^4 -26931977989*n^3 +131963667975*n^2 -188283072995*n +85757456660)*a(n-1) +2*(-61301926003*n^4 +515926265010*n^3 -1655392333929*n^2 +2311146075302*n -1165379619540)*a(n-2) -96*(39221117*n -50949760)*(4*n-9)*(2*n-5)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Aug 19 2025

A386898 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(5*n-k,n-k).

Original entry on oeis.org

1, 11, 199, 4031, 85919, 1885311, 42154111, 955020287, 21847988735, 503573013503, 11675986431999, 272033089535999, 6363380561141759, 149354395882487807, 3515589114309115903, 82957940541503045631, 1961823306198598418431, 46482660516543479939071

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A386812, A386895, A386896, A386897.

Cf. A178792, A383326, A383716.

a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(5*n-k, n-k));

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(4*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n+k,k).
a(n) = binomial(5*n, n)*hypergeom([-1-5*n, -n], [-5*n], -1). - Stefano Spezia, Aug 07 2025

A385639 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(2*n-k,n-k).

Original entry on oeis.org

1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885

Offset: 0

Seiichi Manyama, Aug 07 2025

nonn

Cf. A098430, A383716, A386811.

Cf. A178792, A244038, A386895.

Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025

cp's OEIS Frontend

A386811 a(n) = Sum_{k=0..n} binomial(4*n+1,k).

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Magma

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A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079).

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A383326 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k,n-k).

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A386834 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k-1,n-k).

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A386898 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(5*n-k,n-k).

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A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).

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A383326 a(n) = Sum_{k=0..n} binomial(3n+1,k) binomial(3*n-k,n-k).

A386834 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(4*n-k-1,n-k).

A386898 a(n) = Sum_{k=0..n} binomial(5n+1,k) binomial(5*n-k,n-k).

A385639 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(2*n-k,n-k).