A104859 -id:A104859 - 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

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4k,k) / (3k + 1).

Original entry on oeis.org

1, 2, 7, 38, 257, 1935, 15505, 129519, 1115061, 9823160, 88121887, 802227794, 7392428009, 68819554003, 646276497617, 6114880542117, 58237420303109, 557850829527246, 5370956411708779, 51947475492561014, 504492516832543885, 4917564488572565160

Offset: 0

Ilya Gutkovskiy, Jul 26 2021

nonn

Binomial transform of A002293.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002293, A007317, A104859, A188687, A226974, A346647, A346648, A346649, A346650.

A346646 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[2/3,1,4/3],-256/27) ;
    simplify(%) ;
end proc:
seq(A346646(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Table[Sum[Binomial[n, k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[4 k, k]/(3 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {2/3, 1, 4/3}, -256/27], {n, 0, 21}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)/(3*k + 1)); \\ Michel Marcus, Jul 26 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^2 * A(x)^4.
G.f.: Sum_{k>=0} ( binomial(4*k,k) / (3*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 283^(n + 3/2) / (2048 * sqrt(2*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) -2*(2*n-1) *(91*n^2 -91*n +24)*a(n-1) +6*(n-1) *(155*n^2 -310*n +167)*a(n-2) -438*(n-1) *(n-2)*(2*n-3) *a(n-3) +283*(n-1)*(n-2) *(n-3)*a(n-4)=0. - R. J. Mathar, Aug 17 2023

A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

Original entry on oeis.org

1, 4, 19, 103, 598, 3601, 22165, 138445, 873916, 5560741, 35605756, 229142476, 1480820176, 9603245620, 62463474700, 407330900284, 2662179813931, 17433248900656, 114359597479261, 751343566800961, 4943188072606456

Offset: 0

Emanuele Munarini, Apr 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000(terms 1..100 from Vincenzo Librandi)

Cf. A001764, A005809, A104859, A188676, A188678 - A188687.
Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).
Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).

Table[Sum[Binomial[3k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[3n,n],{n,0,20}]] (* Nearly 300 times faster than the program above. *) (* Harvey P. Dale, Sep 14 2024 *)

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

for(n=0,25, print1(sum(k=0,n, binomial(3*k,k)), ", ")) \\ G. C. Greubel, Jan 27 2017

a(n) = Sum_{k=0..n} binomial(3*k,k).
Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(31*n^2+95*n+72)*a(n+1)+3*(3*n+4)(3*n+5)*a(n)=0.
G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1-x)*sqrt(4-27*x)).
a(n) ~ sqrt(3)*27^(n+1)/(46*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

A188680 Alternate partial sums of binomial(3n,n)^2.

Original entry on oeis.org

1, 8, 217, 6839, 238186, 8779823, 335842273, 13185196127, 527732395714, 21438596184911, 881264330165314, 36575197658193086, 1530121867019096914, 64443673226319500222, 2729760145163758146178, 116203781083772019594878

Offset: 0

Emanuele Munarini, Apr 08 2011

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

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188681, A188682, A188683, A188684, A188685, A188686, A188687.

Cf. Alternate partial sums of binomial(k*n,n)^2: A228002 (k=2), this sequence (k=3).

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

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

a(n)=my(t=1); sum(k=1,n, t*=(27*k^2 - 27*k + 6)/(4*k^2 - 2*k); (-1)^(n-k)*t^2)+(-1)^n \\ Charles R Greathouse IV, Nov 02 2016

a(n) = sum(C(3k,k)^2*(-1)^(n-k), k=0..n).
Recurrence: 4*(2*n^2+7*n+6)^2 * a(n+2) -(713*n^4+4262*n^3+9509*n^2 +9384*n+3456) * 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)/(745*Pi*n*2^(4*n+2)). - Vaclav Kotesovec, Aug 06 2013

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

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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A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) / (3*k + 1).

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A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

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A188680 Alternate partial sums of binomial(3n,n)^2.

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

A346646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4k,k) / (3k + 1).

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