A188680 -id:A188680 - cp's OEIS frontend

A188687 Partial binomial sums of binomial(3n,n)/(2n+1) = A001764(n).

1, 2, 6, 25, 126, 704, 4183, 25897, 165166, 1077520, 7156352, 48222354, 328859011, 2265428728, 15740837575, 110187356134, 776336572878, 5501042194580, 39177463572112, 280277949384146, 2013277273220064, 14514764553512488, 104993261648226446

Offset: 0

Emanuele Munarini, Apr 08 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Cf. A005809, A001764, A188675, A188676, A104859, A188678, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686.

Table[Sum[Binomial[n,k]Binomial[3k,k]/(2k+1),{k,0,n}],{n,0,22}]

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

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(3k,k)/(2k+1).
G.f.: (2/sqrt(3x*(1-x)))*sin((1/3)*arcsin(3/2*sqrt(3*x/(1-x)))).
Recurrence: 2*n*(2*n+1)*a(n) = (39*n^2-35*n+8)*a(n-1) - 2*(n-1)*(33*n-32)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 31^(n+3/2)/(3^4*2^(2*n+2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x) * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021

A188678 Alternate partial sums of binomial(3n,n)/(2n+1).

Original entry on oeis.org

1, 0, 3, 9, 46, 227, 1201, 6551, 36712, 209963, 1220752, 7193888, 42873220, 257957352, 1564809168, 9559946496, 58768808463, 363261736872, 2256369305793, 14076552984507, 88163556913188, 554148894304557, 3494365949734563

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)

Cf. A005809, A001764, A188676, A104859, A188679, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

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

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

a(n) = Sum_{k=0..n} binomial(3*k,k)*(-1)^(n-k)/(2*k+1).
Recurrence: 2*(2*n^2+9*n+10)*a(n+2)-(23*n^2+63*n+40)*a(n+1)-3*(9*n^2+27*n+20)*a(n)=0.
G.f.: 2*sin((1/3)*arcsin(3*sqrt(3*x)/2))/((1+x)*sqrt(3*x)).
a(n) ~ 3^(3*n+3+1/2)/(31*sqrt(Pi)*n^(3/2)*2^(2*n+2)). - Vaclav Kotesovec, Aug 06 2013
G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^2 * A(x)^3. - Ilya Gutkovskiy, Jul 25 2021

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

Original entry on oeis.org

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

A188685 Partial alternating sums of binomial(3n,n)^2/(2n+1)^2.

Original entry on oeis.org

1, 0, 9, 135, 2890, 71639, 1967545, 58125959, 1813561210, 59034994415, 1987910416810, 68818255912790, 2437897047570874, 88061136002276310, 3234416650430634090, 120525771933269446806, 4548292982313797644875

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A001764, A005809, A104859, A188678, A188681.

Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), this sequence (k=2).

[ &+[(-1)^(n-k)*Binomial(3*k, k)^2/(2*k+1)^2: k in [0..n]]: n in [0..16]];  // Bruno Berselli, Apr 11 2011

A001764 := proc(n) binomial(3*n,n)/(2*n+1) ; end proc:
A188685 := proc(n) add( (-1)^(n-k)*A001764(k)^2,k=0..n) ; end proc: # R. J. Mathar, Apr 11 2011

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

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

a(n) = Sum_{k=0..n} (-1)^(n-k)*A001764(k)^2.
4*(2*n^2 + 9*n + 10)^2*a(n+2) - (713*n^4 + 4230*n^3 + 9317*n^2 + 9000*n + 3200)*a(n+1) - 9*(9*n^2 + 27*n + 20)^2*a(n) = 0.
a(n) ~ 3^(6*n+7)/(745*Pi*n^3*2^(4*n+4)). - 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

A188684 Partial sums of binomials binomial(3n,n)^2/(2n+1)^2.

Original entry on oeis.org

1, 2, 11, 155, 3180, 77709, 2116893, 62210397, 1933897566, 62782453191, 2109727864416, 72915894194016, 2579631197677680, 93078664247524864, 3415556450680435264, 127175745034380516160, 4795994499281447607841

Offset: 0

Emanuele Munarini, Apr 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A005809, A001764, A188681.
Cf. Partial sums of binomial(3n,n)^2/(2n+1)^k: A188679 (k=0), A188682 (k=1), this sequence (k=2).
Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), A188685 (k=2).

[&+[Binomial(3*k,k)^2/(2*k+1)^2: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 04 2016

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

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

a(n) = sum( A001764(k)^2 , k=0..n).
4*(2*n^2+9*n+10)^2*a(n+2) - (745*n^4+4518*n^3+10285*n^2+10440*n+4000)*a(n+1) + 9*(9*n^2+27*n+20)^2*a(n) = 0.
a(n) = 4F3(1/3,1/3,2/3,2/3; 1,3/2,3/2; 729/16) - Gamma^2(3n+4) *5F4(1,n+4/3,n+4/3,n+5/3,n+5/3; n+2,n+2,n+5/2,n+5/2; 729/16)/ (Gamma(n+2)*Gamma(2n+3))^2, with pFq() generalized hypergeometric functions. - Charles R Greathouse IV, Apr 14 2011
a(n) ~ 3^(6*n+7)/(713*Pi*n^3*2^(4*n+4)). - Vaclav Kotesovec, Aug 06 2013

A188714 G.f.: (1+x+x^2+x^3)/(1-3x-3x^2-3*x^3).

Original entry on oeis.org

1, 4, 16, 64, 252, 996, 3936, 15552, 61452, 242820, 959472, 3791232, 14980572, 59193828, 233896896, 924213888, 3651913836, 14430073860, 57018604752, 225301777344, 890251367868, 3517715249892, 13899805185312, 54923315409216, 217022507533260, 857536884383364, 3388448121977520, 13389022541682432, 52905022644129948, 209047479923369700

Offset: 0

N. J. A. Sloane, Apr 08 2011

nonn

G.f. for number of ways to spin a dreidel n times without having a run of length 4 of any of gimel, heh, nun or shin.
More generally, fix an alphabet of size M and consider the number of words of length n which do not contain M consecutive equal letters. The present sequence is the case M = 4.
For the cases M=2 through 5 see A040000, A121907, A188714, A188680.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Noonan and D. Zeilberger, The Goulden-Jackson cluster method: extensions, applications and implementations
Doron Zeilberger, Webpage of the paper `The Goulden-Jacskon Cluster Method: Extensions, Applications and Implementations', by John Noonan and Doron Zeilberger; Local copy, pdf file only, no active links
Index entries for linear recurrences with constant coefficients, signature (3, 3, 3).

Cf. A040000, A121907, A188680. Column 4 of A265624.

# First download the Maple package DAVID_IAN from the Zeilberger web site
read(DAVID_IAN);
M:=4;
lis1:={}; for i from 1 to M do lis1:={op(lis1),x[i]}; od:
lis2:={}; for i from 1 to M do t1:=[]; for j from 1 to M do t1:=[op(t1),x[i]]; od: lis2:={op(lis2),t1}; od:
GJs(lis1, lis2, x);

CoefficientList[Series[(1+x+x^2+x^3)/(1-3x-3x^2-3x^3),{x,0,30}], x]  (* Harvey P. Dale, Apr 16 2011 *)

cp's OEIS Frontend

A188687 Partial binomial sums of binomial(3n,n)/(2n+1) = A001764(n).

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A188678 Alternate partial sums of binomial(3*n,n)/(2*n+1).

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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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A188685 Partial alternating sums of binomial(3n,n)^2/(2n+1)^2.

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

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A188678 Alternate partial sums of binomial(3n,n)/(2n+1).

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

A188714 G.f.: (1+x+x^2+x^3)/(1-3x-3x^2-3*x^3).