A383326 -id:A383326 - cp's OEIS frontend

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

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)

A160906 Row sums of A159841.

Original entry on oeis.org

1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912

Offset: 0

R. J. Mathar, May 29 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A075273, A159841.
Cf. A005809, A006256, A183160, A226751.
Cf. A000302, A386811, A386812.
Cf. A244038, A383326.

A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;

Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)

a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017

a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015

a(n) = Sum_{k=0..n} A159841(n,k).
Conjecture: a(2n+1) = A075273(3n).
a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - Peter Luschny, May 19 2015
Conjecture: 2*n*(2*n-1)*(5*n-4)*a(n) +(-295*n^3+451*n^2-130*n-24)*a(n-1) +24*(5*n+1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Jul 20 2016
a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - Ilya Gutkovskiy, Oct 25 2017
a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - Vaclav Kotesovec, Oct 25 2017
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - Seiichi Manyama, Aug 03 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 12 2025
G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g*(6-g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 16 2025

A244038 a(n) = 4^n * binomial(3*n/2,n).

Original entry on oeis.org

1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760

Offset: 0

N. J. A. Sloane, Jun 28 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_1(n).

Cf. A045741, A244039.
Cf. A160906, A383326.
Cf. A000302, A098430, A386897.

[Round(4^n*Gamma(3*n/2+1)/(Gamma(n+1)*Gamma(n/2+1))): n in [0..40]]; // G. C. Greubel, Aug 06 2018

f1:=n->4^n*binomial(3*n/2,n); [seq(f1(n),n=0..40)];

Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* Vincenzo Librandi, Jun 29 2014 *)

{a(n) = if( n<1, n==0, 3^n * polcoeff( serreverse( x / (x+1) / 2 * sqrt((x+3) / (x+1) / 3 + x * O(x^n))), n))}; /* Michael Somos, Jan 27 2018 */

a(n) = A045741(n+1) + A244039(n) [Gessel].
a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - Michael Somos, Jan 27 2018
a(n) = [x^n] ( (1 + 4*x)^(3/2) )^n. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022
G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - Vladimir Kruchinin, Oct 06 2022
G.f.: ((3^(5/6)*i + 3^(1/3))*(-18*i*z + sqrt(-324*z^2 + 3))^(1/3) - (3^(5/6)*i - 3^(1/3))*(18*i*z + sqrt(-324*z^2 + 3))^(1/3))/(2*sqrt(-324*z^2 + 3)), where i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Oct 24 2024
From Seiichi Manyama, Aug 07 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k,n-k). (End)

A383716 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(4*n-k,n-k).

Original entry on oeis.org

1, 9, 127, 2001, 33151, 565249, 9819391, 172826369, 3071424511, 54992986113, 990477877247, 17925526679553, 325710362673151, 5938147061596161, 108571788661555199, 1990032340043366401, 36554697970011340799, 672749920475758460929, 12402180156683794251775

Offset: 0

Seiichi Manyama, Aug 04 2025

nonn

Cf. A178792, A383326.

Cf. A370101.

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

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k).

A386833 a(n) = Sum_{k=0..n} binomial(3n+1,k) binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 6, 59, 656, 7701, 93210, 1150495, 14395428, 181936169, 2317140014, 29691138099, 382334271544, 4943464235069, 64137141682242, 834561532624967, 10886878474010700, 142332442919829585, 1864423992564121686, 24464149489904517211, 321499324010641490016, 4230840338116037836901

Offset: 0

Seiichi Manyama, Aug 05 2025

nonn

Cf. A116881, A386834.
Cf. A370097, A386836.
Cf. A383326.

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

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

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

cp's OEIS Frontend

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

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A160906 Row sums of A159841.

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A244038 a(n) = 4^n * binomial(3*n/2,n).

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A383716 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-k,n-k).

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A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*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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A383716 a(n) = Sum_{k=0..n} binomial(4n+1,k) binomial(4*n-k,n-k).

A386833 a(n) = Sum_{k=0..n} binomial(3n+1,k) binomial(3*n-k-1,n-k).

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