A071879 -id:A071879 - cp's OEIS frontend

A116411 Coefficient of x^n in the expansion of (1+x+x^3)^n.

1, 1, 1, 4, 13, 31, 76, 211, 589, 1597, 4351, 12046, 33496, 93094, 259351, 725089, 2031709, 5701189, 16023181, 45104044, 127134283, 358764613, 1013494318, 2865933907, 8111573416, 22977551656, 65138143006, 184789086106, 524571000799, 1490044262503, 4234901256781, 12042611698876, 34262320521661, 97525522430989

Offset: 0

Paul Barry, Feb 13 2006

Binomial transform of central coefficient of (1+x^3)^n.
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,3). [Joerg Arndt, Jul 05 2011]
a(n) also gives the number of lattice paths from (0,0) to (n,2*n) using steps (1,0), (1,2) and (1,3) since if a*(1,0) + b*(1,1) + c*(1,3) denotes a path from (0,0) to (n,n) then c*(1,0) + b*(1,2) + a*(1,3) is a path from (0,0) to (n,2*n) and vice versa. - Peter Bala, Jul 20 2014

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52.

Cf. A071879.

series(RootOf( (31*x^3-12*x^2+12*x-4)*A^3+(3-3*x)*A+1, A),x=0,30); # Mark van Hoeij, Apr 16 2013

Flatten[{1,Table[Coefficient[Expand[(1+x+x^3)^n],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 11 2015 *)
Table[HypergeometricPFQ[{1/3-n/3,2/3-n/3,-n/3},{1/2,1},-27/4],{n,0,20}] (* Benedict W. J. Irwin, Nov 01 2016 *)

a(n)=polcoeff((1+x+x^3)^n,n); /* Joerg Arndt, Jul 01 2011 */

a(n) = Sum_{k=0..floor(n/3)} C(n, 3*k)*C(3*k, k).
a(n) = Sum_{k=0..n} C(n, k)*C(k, k/3)*(cos(2*Pi*k/3)+1/2)*2/3.
G.f.:  A(x) satisfies (31*x^3-12*x^2+12*x-4)*A(x)^3+(3-3*x)*A(x)+1 = 0.  - Mark van Hoeij, Apr 16 2013
From Peter Bala, Jul 20 2014: (Start)
a(n) = coefficient of x^(2*n) in the expansion of (1 + x^2 + x^3)^n.
O.g.f.: sum {n >= 0} binomial(3*n,n)*x^(3*n)/(1 - x)^(3*n + 1).
By the Lagrange inversion formula, the o.g.f. equals the logarithmic derivative of the o.g.f. of A071879.
E.g.f.: exp(x)*sum {n >= 0} x^(3*n)/((2*n)!*n!). (End)
Recurrence: 2*n*(2*n-3)*a(n) = 2*(6*n^2 - 12*n + 5)*a(n-1) - 6*(n-1)*(2*n-3)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Feb 11 2015
a(n) ~ (3 + 2^(2/3))^(n+1/2) / (2^(2*n/3 + 1) * sqrt(3*Pi*n)). - Vaclav Kotesovec, Feb 11 2015
a(n) = 3F2[1/3-n/3,2/3-n/3,-n/3; 1/2,1; -27/4]. - Benedict W. J. Irwin, Nov 01 2016

A364522 G.f. satisfies A(x) = 1 + xA(x) + x^5A(x)^5.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 258, 518, 1123, 2718, 7008, 18054, 44969, 108189, 255919, 609179, 1482210, 3689155, 9294440, 23419705, 58639835, 145948111, 362721386, 904673836, 2270287636, 5729191861, 14502873988, 36735974548, 93001413353, 235372519273

Offset: 0

Seiichi Manyama, Jul 27 2023

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 5. - Emanuele Munarini, Jul 11 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001006, A071879, A127902, A364523.

Cf. A182454, A349311.

a(n) = sum(k=0, n\5, binomial(n, 5*k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..floor(n/5)} binomial(n,5*k) * binomial(5*k,k) / (4*k+1).

A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 931, 1795, 3550, 7736, 18929, 49505, 130000, 330430, 804271, 1885675, 4327555, 9929515, 23224435, 55907251, 138016906, 345107296, 862546231, 2136402451, 5231163232, 12697101118, 30723857209, 74569942745

Offset: 0

Seiichi Manyama, Jul 27 2023

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 6. - Emanuele Munarini, Jul 11 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001006, A071879, A127902, A364522.

Cf. A349312, A364472.

a(n) = sum(k=0, n\6, binomial(n, 6*k)*binomial(6*k, k)/(5*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * binomial(6*k,k) / (5*k+1).

A369483 Expansion of (1/x) * Series_Reversion( x / (1+x+x^3)^2 ).

Original entry on oeis.org

1, 2, 5, 16, 60, 242, 1014, 4370, 19278, 86678, 395751, 1829742, 8549100, 40302810, 191469165, 915751966, 4405727502, 21307102900, 103526683797, 505118705078, 2473833623696, 12157124607612, 59929746189165, 296271556144028, 1468494529164194, 7296261411708962

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A369484, A369485.

Cf. A071879.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^3)^2)/x)

a(n, s=3, t=2, u=0) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

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

A366051 Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).

Original entry on oeis.org

1, -1, 1, 0, -3, 9, -16, 13, 29, -157, 391, -562, -32, 3002, -10373, 20747, -18083, -47941, 271117, -712216, 1066699, 122131, -6464446, 22907125, -46951992, 40883304, 120187926, -679375906, 1809757015, -2731745887, -468147579, 17768126376, -63256877763

Offset: 0

Seiichi Manyama, Sep 27 2023

sign

Cf. A049128, A114997, A366052, A366053.

Cf. A071879.

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

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

A367113 G.f. satisfies A(x) = 1 + 2xA(x) + 2x^3A(x)^3.

Original entry on oeis.org

1, 2, 4, 10, 32, 112, 396, 1416, 5184, 19424, 73984, 285056, 1108848, 4350816, 17203008, 68473504, 274122752, 1103014912, 4458611968, 18096793088, 73724852224, 301360575488, 1235633545216, 5080554352640, 20943623880448, 86541514460672, 358386391051264

Offset: 0

Seiichi Manyama, Nov 05 2023

Cf. A001764, A071879, A367111, A367112.
Cf. A071356, A367115.
Cf. A367074.

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

a(n) = Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(n,3*k) * A001764(k).

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 39, 78, 169, 373, 808, 1727, 3719, 8153, 18100, 40315, 89770, 200250, 448755, 1010685, 2284295, 5173961, 11740697, 26699780, 60863291, 139045991, 318247190, 729572315, 1675085099, 3851795549, 8869990949, 20453679944, 47223844863

Offset: 0

Seiichi Manyama, Jul 28 2023

Cf. A000108, A071879, A346626.

Cf. A023426, A063019, A127902.

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

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

A365732 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 57, 88, 142, 250, 473, 917, 1751, 3240, 5829, 10350, 18472, 33574, 62293, 117138, 220932, 414777, 773282, 1434776, 2661302, 4955167, 9279325, 17466103, 32971057, 62274094, 117521503, 221572762, 417699772, 788205724, 1489975777

Offset: 0

Seiichi Manyama, Sep 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A071879, A364552, A365252.

Cf. A001764, A365734.

a(n) = sum(k=0, n\6, binomial(n-5*k, k)*binomial(n-3*k+1, n-5*k)/(n-3*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n-5*k,k) * binomial(n-3*k+1,n-5*k) / (n-3*k+1) = Sum_{k=0..floor(n/6)} binomial(n-3*k,3*k) * binomial(3*k,k) / (2*k+1).

A367111 G.f. satisfies A(x) = 1 + xA(x) + 2x^3*A(x)^3.

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 53, 155, 449, 1273, 3721, 11155, 33529, 101245, 309037, 950587, 2936833, 9117169, 28448209, 89134435, 280252585, 884123429, 2797933733, 8879167067, 28249550913, 90091462761, 287946752601, 922194331891, 2959055180953, 9511538457229

Offset: 0

Seiichi Manyama, Nov 05 2023

Cf. A001764, A071879, A367112, A367113.

Cf. A190590.

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

a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(n,3*k) * A001764(k).

A367112 G.f. satisfies A(x) = 1 + 2xA(x) + x^3*A(x)^3.

Original entry on oeis.org

1, 2, 4, 9, 24, 72, 227, 730, 2384, 7916, 26704, 91280, 315319, 1098710, 3856948, 13628441, 48435808, 173030048, 620965396, 2237681720, 8093572960, 29372735368, 106925552672, 390336084256, 1428620011263, 5241166583502, 19270575881964

Offset: 0

Seiichi Manyama, Nov 05 2023

Cf. A001764, A071879, A367111, A367113.

Cf. A367073.

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

a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(n,3*k) * A001764(k).

cp's OEIS Frontend

A116411 Coefficient of x^n in the expansion of (1+x+x^3)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A364522 G.f. satisfies A(x) = 1 + x*A(x) + x^5*A(x)^5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A364523 G.f. satisfies A(x) = 1 + x*A(x) + x^6*A(x)^6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369483 Expansion of (1/x) * Series_Reversion( x / (1+x+x^3)^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A366051 Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A367113 G.f. satisfies A(x) = 1 + 2*x*A(x) + 2*x^3*A(x)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A364552 G.f. satisfies A(x) = 1 + x*A(x) + x^4*A(x)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365732 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^5*A(x)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A367111 G.f. satisfies A(x) = 1 + x*A(x) + 2*x^3*A(x)^3.

Views

Author

Keywords

A364522 G.f. satisfies A(x) = 1 + xA(x) + x^5A(x)^5.

A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.

A367113 G.f. satisfies A(x) = 1 + 2xA(x) + 2x^3A(x)^3.

A364552 G.f. satisfies A(x) = 1 + xA(x) + x^4A(x)^3.

A365732 G.f. satisfies A(x) = 1 + xA(x)(1 + x^5*A(x)^2).

A367111 G.f. satisfies A(x) = 1 + xA(x) + 2x^3*A(x)^3.

A367112 G.f. satisfies A(x) = 1 + 2xA(x) + x^3*A(x)^3.