A062236 -id:A062236 - cp's OEIS frontend

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821

Offset: 0

N. J. A. Sloane

nonn

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006256.
Cf. A001764, A006013, A007318, A036917, A062236.
Cf. A000302, A308523, A386367, A386368.

a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012

Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - Seiichi Manyama, Jul 19 2025
G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025

A386617 a(n) = Sum_{k=0..n-1} binomial(3k+1,k) binomial(3n-3k,n-k-1).

Original entry on oeis.org

0, 1, 10, 81, 610, 4436, 31626, 222681, 1554772, 10790721, 74560728, 513452604, 3526463304, 24168921568, 165357919850, 1129724254953, 7709039995368, 52551835079699, 357930487932282, 2436038623348521, 16568626556643738, 112626521811112464, 765201654587796312, 5196570956399432796

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A006256, A036829, A062236, A075045.

Cf. A006419, A386612, A386614, A386616.

a(n) = sum(k=0, n-1, binomial(3*k+1, k)*binomial(3*n-3*k, n-k-1));

G.f.: g^3 * (g-1)/(3-2*g)^2 where g=1+x*g^3.
G.f.: g/((1-g)^2 * (1-3*g)^2) where g*(1-g)^2 = x.
a(n) = Sum_{k=0..n-1} binomial(3*k+1+l,k) * binomial(3*n-3*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(3*n+2,k).
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(2*n+k+2,k).

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 11, 111, 1091, 10596, 102237, 982458, 9415539, 90063180, 860278156, 8208539351, 78258171957, 745595635084, 7099714918062, 67574576298276, 642927956583123, 6115089154367484, 58146652079312580, 552769690436583532, 5253812277363417836, 49925987913040522128

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...

Cf. A000346, A062236, A386566, A386567.

Cf. A002293, A006632, A308523.

a(n) = sum(k=0, n-1, binomial(4*k-1, k)*binomial(4*n-4*k, n-k-1));

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))

G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(4*k-1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,k).

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 14, 181, 2284, 28506, 353630, 4370584, 53882392, 663116347, 8150224204, 100073884670, 1227826127020, 15055154471696, 184508186225552, 2260299193652496, 27679951219660080, 338872887728053465, 4147618793911034330, 50753529798492061819, 620942367878256638264

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ) = x + 7*x^2 + 181*x^3/3 + 571*x^4 + 28506*x^5/5 + ...

Cf. A000346, A062236, A386565, A386567.
Cf. A079678, A386367, A386613, A386614.
Cf. A002294, A118971.

a(n) = sum(k=0, n-1, binomial(5*k-1, k)*binomial(5*n-5*k, n-k-1));

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(5*k, k)/(4*k+1)*x^k)); concat(0, Vec(g*(g-1)/(5-4*g)^2))

G.f.: g*(g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(5*k-1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k,k).
Conjecture D-finite with recurrence 196608*n*(4*n-3)*(2*n-1)*(18270873280*n -32560150837) *(4*n-1)*a(n) +1280*(-1399185802400000*n^5 +1022280893000000*n^4 +17669158913120000*n^3 -48968110172924750*n^2 +49502057719349955*n -17877514345852392)*a(n-1) +125000*(-61298198200000*n^5 +1447969779032500*n^4 -7721498995066250*n^3 +17474948768595875*n^2 -18352567310653770*n +7399184154389181)*a(n-2) +48828125*(5*n-11) *(5*n-14)*(4958243695*n -6717884799) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

Original entry on oeis.org

0, 1, 17, 268, 4129, 62955, 954392, 14417376, 217279857, 3269099590, 49125066135, 737516631908, 11064270530632, 165889863957065, 2486052264852180, 37241727274394640, 557707191712371729, 8349517132932620730, 124971965902300790390, 1870139909398530770760

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...

Cf. A000346, A062236, A386565, A386566.
Cf. A079679, A386368, A386615, A386616.
Cf. A002295, A130564.

a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));

my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))

G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).

A075045 Coefficients A_n for the s=3 tennis ball problem.

Original entry on oeis.org

1, 9, 69, 502, 3564, 24960, 173325, 1196748, 8229849, 56427177, 386011116, 2635972920, 17974898872, 122430895956, 833108684637, 5664553564440, 38488954887171, 261369752763963, 1774016418598269, 12035694958994142, 81624256468292016, 553377268856455968

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

T. Amdeberhan, Integrality of a sum.
Roland Bacher, On generating series of complementary plane trees, arXiv:math/0409050 [math.CO], 2004.
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), pp. 307-344 (A_n for s=3).

See A049235 for more information.

Cf. A006256, A036829, A062236, A386617.

FussArea := proc(s,n)
    local a,i,j ;
    a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
    add(j *(n-j) *binomial((s+1)*j,j) *binomial((s+1)*(n-j),n-j) /(s*j+1) /(s*(n-j)+1),j=0..n) ;
    a := a+binomial(s+1,2)*% ;
    for j from 0 to n-1 do
        for i from 0 to j do
            i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
            *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
            *binomial((s+1)*(n-j)-2,n-1-j) ;
            a := a-%*binomial(s+1,2) ;
        end do:
    end do:
    a ;
end proc:
seq(FussArea(2,n),n=1..30) ; # R. J. Mathar, Mar 31 2023

FussArea[s_, n_] := Module[{a, i, j, pc}, a = Binomial[(s + 1)*n, n]*n/(s*n + 1); pc = Sum[j*(n - j)*Binomial[(s + 1)*j, j]*Binomial[(s + 1)*(n - j), n - j]/(s*j + 1)/(s*(n - j) + 1), {j, 0, n}]; a = a + Binomial[s + 1, 2]*pc; For[j = 0, j <= n - 1 , j++, For[i = 0, i <= j, i++, pc = i*(j - i)/(s*i + 1)/(s*(j - i) + 1)/(n - j)*Binomial[(s + 1)*i, i]* Binomial[(s + 1)*(j - i), j - i]*Binomial[(s + 1)*(n - j) - 2, n - 1 - j]; a = a - pc*Binomial[s + 1, 2]; ]]; a];
Table[FussArea[2, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

G.f.: seems to be (3*g-1)^(-2)*(1-g)^(-3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
Conjecture: D-finite with recurrence 8*(2*n+3)*(7*n+1)*(n+1)*a(n) +6*(-252*n^3-477*n^2-220*n-11)*a(n-1) +81*(7*n+8)*(3*n-1)*(3*n+1)*a(n-2)=0. - Jean-François Alcover, Feb 07 2019
a(n) = (3n+2)*(n+1)*binomial(3n+3,n+1)/2/(2n+3) - A049235(n). [Merlini Theorem 2.5 for s=3] - R. J. Mathar, Oct 01 2021
From Seiichi Manyama, Jul 28 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*k+3+l,k) * binomial(3*n-3*k-l,n-k) for every real number l.
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+4,k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k+3,k). (End)

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)(3n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).

Original entry on oeis.org

1, 5, -6, 16, -60, 48, 44, -288, 660, -440, 112, -1056, 4032, -7280, 4368, 272, -3360, 17952, -52224, 81600, -45696, 640, -9792, 67200, -267520, 656640, -930240, 496128, 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664

Offset: 1

Peter Luschny, Oct 28 2022

			[1]    1;
[2]    5,     -6;
[3]   16,    -60,     48;
[4]   44,   -288,    660,     -440;
[5]  112,  -1056,   4032,    -7280,    4368;
[6]  272,  -3360,  17952,   -52224,   81600,   -45696;
[7]  640,  -9792,  67200,  -267520,  656640,  -930240,   496128;
[8] 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664;

Cf. A062236, A000309.

def P(n):
    h = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x)
    return h.series(x, n+1).polynomial(SR)
for n in range(1, 9): print(P(n).list())
# To evaluate the polynomials use:
def p(n, t): return Integer(P(n)(x=t).n())
# For example the next statements yield A062236 and A000309.
print([p(n, -1/2) for n in range(1, 21)])
print([(-1)^n*p(n + 1, 1) for n in range(0, 22)])

P(n, -1/2) = A062236(n).

(-1)^n*P(n + 1, 1) = A000309(n).

A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 3, 6, 1, 0, 0, 4, 10, 8, 1, 0, 0, 0, 12, 21, 10, 1, 0, 0, 0, 12, 32, 36, 12, 1, 0, 0, 0, 6, 45, 72, 55, 14, 1, 0, 0, 0, 8, 36, 119, 140, 78, 16, 1, 0, 0, 0, 0, 46, 144, 270, 244, 105, 18, 1, 0, 0, 0, 0, 32, 164, 416, 550, 392, 136, 20, 1

Offset: 1

N. J. A. Sloane, Jan 17 2012

The number of nodes is k + 1. The path-length is the sum of the distances of all nodes from the root node. - Andrew Howroyd, Nov 19 2024

			Triangle begins:
1
0 1
0 2 1
0 0 4 1
0 0 3 6 1
0 0 4 10 8 1
0 0 0 12 21 10 1
0 0 0 12 32 36 12 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87 (see Th. 6).

Row sums are A132332.
Column sums are A001764.
Cf. A062236.

T(n)={my(g=1+O(x)); for(i=1, n, g=1/(1 - x*y*subst(g,y,x*y)^2)); [Vecrev(p/y) | p<-Vec(g-1)]}
{my(A=T(10)); for(i=1, #A, print(A[i]))} \\ Andrew Howroyd, Nov 19 2024

From Andrew Howroyd, Nov 19 2024: (Start)
G.f.: A(x,y) satisfies A(x,y) = 1/(1 - x*y*A(x,x*y)^2).
T(k*(k+1)/2, k) = 2^(k-1).
T(n,k) = 0 for n > k*(k+1)/2.
Sum_{n>=1} n*T(n,k) = A062236(k). (End)

a(34) corrected and a(42) onwards from Andrew Howroyd, Nov 19 2024

cp's OEIS Frontend

A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A386617 a(n) = Sum_{k=0..n-1} binomial(3*k+1,k) * binomial(3*n-3*k,n-k-1).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

Formula

A386566 a(n) = Sum_{k=0..n-1} binomial(5*k-1,k) * binomial(5*n-5*k,n-k-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

Formula

A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

Formula

A075045 Coefficients A_n for the s=3 tennis ball problem.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)*(3*n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).

Views

Author

Keywords

Examples

Crossrefs

Programs

SageMath

Formula

A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

A386617 a(n) = Sum_{k=0..n-1} binomial(3k+1,k) binomial(3n-3k,n-k-1).

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)(3n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).