A188681 -id:A188681 - 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

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

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

A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 10, 5, 14, 35, 45, 35, 14, 42, 126, 196, 196, 126, 42, 132, 462, 840, 1008, 840, 462, 132, 429, 1716, 3564, 4950, 4950, 3564, 1716, 429, 1430, 6435, 15015, 23595, 27225, 23595, 15015, 6435, 1430

Offset: 0

F. Chapoton, Mar 21 2024

The terms can be seen as graded dimensions of a non-symmetric operad. The Koszul dual operad has Hilbert series x*(1 + x)*(1 + tx). So the current table has as Hilbert series the reverse of x*(1-x)*(1-t*x) w.r.t to x (see Sage below).

The triangle is symmetric under the exchange of k with n - k.

			Triangle begins:
  [0] [ 1],
  [1] [ 1,   1],
  [2] [ 2,   3,   2],
  [3] [ 5,  10,  10,   5],
  [4] [14,  35,  45,  35,  14],
  [5] [42, 126, 196, 196, 126, 42].

Column 0 and main diagonal are A000108.
Column 1 and subdiagonal are A001700.
Row sums are A006013.
The even bisection of the alternating row sums is A001764.
The central terms are A188681.
Cf. A085614, A250886, A371400.

T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);  # Peter Luschny, Mar 21 2024

T[n_, k_] := (Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, k - n, 1, 1]) /(n + 1); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Peter Luschny, Mar 21 2024 *)

def Trow(n):
    return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]

# As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
t = polygen(QQ, 't')
x = LazyPowerSeriesRing(t.parent(), 'x').0
gf = x*(1-x)*(1-t*x)
coeffs = gf.revert() / x
for n in range(6):
    print(coeffs[n].list())

From Peter Luschny, Mar 21 2024: (Start)
T(n, k) = hypergeom([-n, -k], [1], 1)*hypergeom([-n, k - n], [1], 1)/(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1). (End)

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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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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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).