A188678 -id:A188678 - cp's OEIS frontend

A188682 Partial sums of binomials bin(3n,n)^2/(2n+1).

1, 4, 49, 1057, 28282, 848101, 27357493, 928760053, 32747441926, 1188869998801, 44174723634526, 1672716549215326, 64340599136306926, 2507814491482180894, 98859670298036582494, 3935425516392739090270, 158006444406545953115743

Offset: 0

Emanuele Munarini, Apr 08 2011

Harvey P. Dale, Table of n, a(n) for n = 0..606

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188681, A188683, A188684, A188685, A188686, A188687.

Table[Sum[Binomial[3k,k]^2/(2k+1),{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2/(2n+1),{n,0,20}]] (* Harvey P. Dale, Jul 10 2016 *)

makelist(sum(binomial(3*k,k)^2/(2*k+1),k,0,n),n,0,20);

a(n) = sum(bin(3*k,k)^2/(2*k+1),k=0..n).
Recurrence: 4*(n+2)^2*(4*n^2+16*n+15) * a(n+2) -(745*n^4+4502*n^3+10181*n^2+10216*n+3840) * a(n+1) +9*(9*n^2+27*n+20)^2 *a(n) = 0.
a(n) ~ 3^(6*n+7)/(713*Pi*n^2*2^(4*n+3)). - Vaclav Kotesovec, Aug 06 2013

A346628 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x) * A(x)^3.

Original entry on oeis.org

1, 0, 2, 5, 22, 92, 415, 1927, 9198, 44804, 221880, 1113730, 5653747, 28975962, 149725355, 779178092, 4080167790, 21483383992, 113670233848, 604070682354, 3222823434608, 17255628041720, 92689459311470, 499359484166994, 2697571066055611, 14608820993453132

Offset: 0

Ilya Gutkovskiy, Jul 25 2021

nonn

Inverse binomial transform of A001764.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001764, A005043, A188678, A188687, A346627, A346664, A346665, A346666, A346667, A346668.

nmax = 25; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x) A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 25; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 25}]

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(3*k,k) / (2*k + 1).
a(n) ~ 23^(n + 3/2) / (81 * sqrt(Pi) * n^(3/2) * 2^(2*n+2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence +2*n*(2*n+1)*a(n) -(15*n-4)*(n-1)*a(n-1) -2*(n-1)*(21*n-22)*a(n-2) -23*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Aug 05 2021

A346680 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4k,k) / (3k + 1).

Original entry on oeis.org

1, 0, 4, 18, 122, 847, 6237, 47583, 373149, 2989111, 24354777, 201214021, 1681719343, 14193619647, 120800146953, 1035593096367, 8934344395053, 77510878324671, 675799844685937, 5918354494345863, 52037647837001257, 459200394617540288, 4065477723321641932

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A002293, A032357, A188678, A345367, A346664.

Cf. A346681, A346682, A346683, A346684.

Table[Sum[(-1)^(n - k) Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^3 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(4*k, k)/(3*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^3 * A(x)^4.
a(n) ~ 2^(8*n + 17/2) / (283 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -(n-1)*(229*n^2-155*n+24)*a(n-1) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Aug 05 2021

A346681 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5k,k) / (4k + 1).

Original entry on oeis.org

1, 0, 5, 30, 255, 2275, 21476, 210404, 2120041, 21830314, 228713056, 2430255074, 26128088701, 283703487059, 3106713300821, 34270543858459, 380471319687826, 4247891403168599, 47665096853113576, 537244509843680309, 6079834137116933061, 69054467456964456599

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..925

Cf. A002294, A032357, A188678, A345368, A346665.

Cf. A346680, A346682, A346683, A346684.

Table[Sum[(-1)^(n - k) Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^4 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(5*k, k)/(4*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^4 * A(x)^5.

a(n) ~ 5^(5*n + 11/2) / (3381 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021

A346682 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6k,k) / (5k + 1).

Original entry on oeis.org

1, 0, 6, 45, 461, 5020, 57812, 691586, 8512048, 107095262, 1371219004, 17808830924, 234048288772, 3106795261083, 41593689788637, 560980967638479, 7614970691479315, 103957059568762775, 1426355910771621805, 19658792867492660060, 272046427837226505466

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..856

Cf. A002295, A032357, A188678, A346065, A346666.

Cf. A346680, A346681, A346683, A346684.

Table[Sum[(-1)^(n - k) Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(6*k, k)/(5*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^5 * A(x)^6.

a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (49781 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 0, 7, 63, 756, 9716, 132062, 1865626, 27124049, 403197584, 6100155272, 93626517858, 1454221328232, 22815183746508, 361030984965596, 5755543515895284, 92350704790963431, 1490287557170676816, 24171116970619575559, 393808998160695560841, 6442255541764422795759

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A002296, A032357, A188678, A346667, A346671.

Cf. A346680, A346681, A346682, A346684.

Table[Sum[(-1)^(n - k) Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^6 * A(x)^7.

a(n) ~ 7^(7*n + 15/2) / (870199 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8k,k) / (7k + 1).

Original entry on oeis.org

1, 0, 8, 84, 1156, 17122, 268262, 4370086, 73281938, 1256608767, 21933420953, 388400019583, 6960642974905, 126008367913375, 2300862338502425, 42326714610861679, 783717720798538121, 14594469249932149279, 273161824453612674593, 5135931850101477641707

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m + (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Seiichi Manyama, Table of n, a(n) for n = 0..768

Cf. A007556, A032357, A188678, A346668, A346672.

Cf. A346680, A346681, A346682, A346683.

Table[Sum[(-1)^(n - k) Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Jul 29 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^7 * A(x)^8.

a(n) ~ 2^(24*n + 25) / (17600759 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

cp's OEIS Frontend

A188682 Partial sums of binomials bin(3n,n)^2/(2n+1).

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A346628 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x) * A(x)^3.

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A346680 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*k,k) / (3*k + 1).

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A346681 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*k,k) / (4*k + 1).

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A346682 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6*k,k) / (5*k + 1).

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A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7*k,k) / (6*k + 1).

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A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*k,k) / (7*k + 1).

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A346680 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4k,k) / (3k + 1).

A346681 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5k,k) / (4k + 1).

A346682 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(6k,k) / (5k + 1).

A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7k,k) / (6k + 1).

A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8k,k) / (7k + 1).