A188687 -id:A188687 - cp's OEIS frontend

A188676 Alternate partial sums of the binomial coefficients binomial(3*n,n).

1, 2, 13, 71, 424, 2579, 15985, 100295, 635176, 4051649, 25993366, 167543354, 1084134346, 7038291098, 45821937982, 299045487602, 1955803426045, 12815265660680, 84111082917925, 552872886403775, 3638971619401720

Offset: 0

Emanuele Munarini, Apr 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A005809, A001764, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Table[Sum[Binomial[3k, k](-1)^(n-k), {k, 0, n}], {n, 0, 20}]

makelist(sum(binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,20);

a(n) = sum(k=0..n, (-1)^(n-k)*binomial(3*k,k) ).
Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(23*n^2+67*n+48)*a(n+1)-3*(3*n+4)*(3n+5)*a(n)=0.
G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(4-27*x)).
a(n) ~ 3^(3*n+7/2)/(62*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

A188681 a(n) = binomial(3n,n)^2/(2n+1).

Original entry on oeis.org

1, 3, 45, 1008, 27225, 819819, 26509392, 901402560, 31818681873, 1156122556875, 42985853635725, 1628541825580800, 62667882587091600, 2443473892345873968, 96351855806554401600, 3836565846094702507776, 154071018890153214025473

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..606
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188682, A188683, A188684, A188685, A188686, A188687.

Table[Binomial[3k,k]^2/(2k+1),{k,0,20}]
CoefficientList[Series[HypergeometricPFQ[{1/3,1/3,2/3,2/3}, {1/2,1,3/2}, (729 x)/16],{x,0,20}],x]  (* Harvey P. Dale, Apr 22 2011 *)

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

Recurrence: 4*(n+1)^2*(2*n+1)*(2*n+3)*a(n+1)-9*(3*n+1)^2*(3*n+2)^2*a(n)=0.

a(n) ~ 3^(6*n+1)/(Pi*2^(4*n+3)*n^2). - Vaclav Kotesovec, Aug 16 2013

A188679 Partial sums of binomial(3n,n)^2.

Original entry on oeis.org

1, 10, 235, 7291, 252316, 9270325, 353892421, 13874930821, 554792522662, 22521121103287, 925224047453512, 38381686035811912, 1605078750713101912, 67578873844051699048, 2861782692234129345448, 121795323921169907086504

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..604

Cf. A005809, A001764, A188676, A104859, A188678, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Table[Sum[Binomial[3k,k]^2,{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2,{n,0,20}]] (* Harvey P. Dale, Sep 26 2019 *)

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

a(n) = sum(C(3k,k)^2, k=0..n).
Recurrence: 4*(2*n^2+7*n+6)^2 * a(n+2) -(745*n^4+4486*n^3+10093*n^2 +10056*n+3744) * a(n+1) +9*(9*n^2+27*n+20)^2 * a(n) = 0.
G.f.: (1-x)^(-1)*F(1/3,1/3,2/3,2/3;1/2,1/2,1;729*x/16), where F(a1,a2,a3,a4;b1,b2,b3;z) is a hypergeometric series.
a(n) ~ 3^(6*n+7)/(713*Pi*n*2^(4*n+2)). - Vaclav Kotesovec, Aug 06 2013

A188683 Alternate partial sums of binomial(3n,n)^2/(2n+1).

Original entry on oeis.org

1, 2, 43, 965, 26260, 793559, 25715833, 875686727, 30942995146, 1125179561729, 41860674073996, 1586681151506804, 61081201435584796, 2382392690910289172, 93969463115644112428, 3742596382979058395348

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..606

Cf. A005809, A001764, A188676, A104859, A188678, A188681, A188686, A188687.
Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), this sequence (k=1), A188685 (k=2).
Cf. Partial sums of binomial(3n,n)^2/(2n+1)^k: A188679 (k=0), A188682 (k=1), A188684 (k=2).

Table[Sum[Binomial[3k,k]^2(-1)^(n-k)/(2k+1),{k,0,n}],{n,0,20}]

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

a(n) = sum(binomial(3*k,k)^2*(-1)^(n-k)/(2*k+1), k=0..n).
Recurrence: 4*(n+2)^2*(4*n^2+16*n+15) * a(n+2) -(713*n^4+4246*n^3 +9421*n^2 +9224*n+3360) * a(n+1) -9*(9*n^2+27*n+20)^2 * a(n) = 0.
a(n) ~ 3^(6*n+7)/(745*Pi*n^2*2^(4*n+3)). - Vaclav Kotesovec, Aug 06 2013

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

Original entry on oeis.org

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

A364592 G.f. satisfies A(x) = 1/(1-x) + x(1-x)A(x)^4.

Original entry on oeis.org

1, 2, 8, 49, 365, 3001, 26193, 238119, 2230151, 21368167, 208459419, 2063563791, 20675793627, 209277092776, 2136720896514, 21979879393677, 227582114799201, 2369983696546858, 24806423607475896, 260829829404493787, 2753744691645428399

Offset: 0

Seiichi Manyama, Jul 29 2023

Cf. A014137, A188687.

Cf. A364594, A364596.

a(n) = sum(k=0, n, binomial(n+k, 2*k)*binomial(4*k, k)/(3*k+1));

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

A346762 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x) A(x)^3.

Original entry on oeis.org

1, 3, 11, 50, 271, 1655, 10900, 75388, 539295, 3954593, 29557251, 224308078, 1723659436, 13384272660, 104855628776, 827760536528, 6578127170319, 52581460222645, 422478996770305, 3410174204693310, 27640220748529799, 224866485110361767, 1835589569664256976

Offset: 0

Ilya Gutkovskiy, Aug 02 2021

nonn

Second binomial transform of A001764.

Cf. A001764, A064613, A188687, A346763.

nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[2^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -27/8], {n, 0, 22}]

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

a(n) ~ 35^(n + 3/2) / (81 * sqrt(Pi) * n^(3/2) * 4^(n+1)). - Vaclav Kotesovec, Nov 26 2021

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).

Original entry on oeis.org

1, 2, 5, 25, 257, 4361, 104425, 3241316, 123865313, 5628753361, 296671566941, 17798975341467, 1197924420178381, 89394126594968755, 7326377073291002147, 654215578855903951141, 63225054646397348577601, 6575059243843086616460321, 732138834180570978286488133

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A007317, A007556, A188687, A346646, A346647, A346648, A346649, A346650.

Cf. A378326.

Table[Sum[Binomial[n, k] Binomial[n*k, k]/((n-1)*k + 1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).

Original entry on oeis.org

1, 2, 10, 65, 480, 3824, 32039, 278256, 2482578, 22617830, 209540672, 1968031520, 18696064179, 179332892186, 1734451272240, 16895744042472, 165621305486976, 1632518433458400, 16170959983623314, 160888256475481560, 1607061512154585046, 16110030923830784248

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A188687, A381817, A381829.
Cf. A129442, A381831.
Cf. A000108, A368975.

my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^2)/x)^(1/2))

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*n-2*k,n-k)/(2*n+k+1).
D-finite with recurrence +432*n*(n-1)*(n-2)*(2*n+1)*(2*n-1)*(2*n-3)*(262261060139434887136491*n -880264534325728808928710)*a(n) +24*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(9441398165019655936913676*n^3 -1563359509176097527827297363*n^2 +8122005300033248841454135898*n -10005843136737488906545668303)*a(n-1) -8*(n-2)*(2*n-3)*(26904862014415612504704360259*n^5 -439294650192331167438487778367*n^4 +2462557164881954865201862193560*n^3 -6116391863054255517662202621591*n^2 +6730597164009721987374566778403*n -2508886036978141982914230533400)*a(n-2) +2*(3280856375160701992555505608813*n^7 -60505233834440544774094319915261*n^6 +458650706405377012453301766859297*n^5 -1843996542698657351167896639498197*n^4 +4199211312282774397146042070543498*n^3 -5283107978583820687249123910721062*n^2 +3195330463869279708956264243293272*n -571272270914692694572799416918200)*a(n-3) +3*(-10499174187769013704183946812135*n^7 +189831332911960443054698384732480*n^6 -1395267797131742288585801071743534*n^5 +5221938509132769354051685228032464*n^4 -9839826026184653630837080778918103*n^3 +6229383740555425356174546560814416*n^2 +6216439623275682391743799709941612*n -8390747283534155728971424365124320)*a(n-4) -112*(7*n-31)*(7*n-32) *(2094251874056865218841652*n -5622141652266976856940223)*(7*n-29)*(7*n-26) *(7*n-30)*(7*n-27)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

cp's OEIS Frontend

A188676 Alternate partial sums of the binomial coefficients binomial(3*n,n).

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A188681 a(n) = binomial(3*n,n)^2/(2*n+1).

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A188679 Partial sums of binomial(3n,n)^2.

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A188683 Alternate partial sums of binomial(3n,n)^2/(2n+1).

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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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A364592 G.f. satisfies A(x) = 1/(1-x) + x*(1-x)*A(x)^4.

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A346762 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x) * A(x)^3.

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A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) / ((n-1)*k + 1).

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A188681 a(n) = binomial(3n,n)^2/(2n+1).

A364592 G.f. satisfies A(x) = 1/(1-x) + x(1-x)A(x)^4.

A346762 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x) A(x)^3.

A378327 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(nk,k) / ((n-1)k + 1).