A116411 -id:A116411 - cp's OEIS frontend

A071879 G.f. satisfies: A(x) = 1 + xA(x) + x^3A(x)^3.

1, 1, 1, 2, 5, 11, 24, 57, 141, 349, 871, 2212, 5688, 14730, 38403, 100829, 266333, 706997, 1885165, 5047522, 13565203, 36578497, 98934826, 268342933, 729709432, 1989021256, 5433518806, 14873285506, 40790118487, 112064912455, 308390452661, 849969894794, 2346045295997

Offset: 0

Paul D. Hanna, Jun 10 2002

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 3. - Emeric Deutsch, Nov 03 2002. [Comment corrected by Christian G. Bower, Sep 25 2007]
Sequence is a Motzkin-like sequence. The Motzkin sequence A001006 counts ordered trees with n edges and having nodes of outdegree 0, 1, or 2 [g.f. f(x) defined by f = 1+x*f+(x*f)^2]. - Emeric Deutsch, Sep 30 2007
G.f. (offset 1) is series reversion of x^2/(x+x^2+x^4).

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 24*x^6 + ...
The first-order differences of the terms form the coefficients of A(x)^3:
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 208*x^6 + 522*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Olivier Bodini, Matthieu Dien, Antoine Genitrini, and Frédéric Peschanski, The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes. In 2017 Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO).
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A001006, A001764, A116411.

a:= n-> add(binomial(n+1,1+2*i)*binomial(n-2*i,i), i=0..floor(n/3))/(n+1): seq(a(n), n=0..29);

a[n_] := Sum[Binomial[n+1, 1+2i]*Binomial[n-2i, i], {i, 0, Floor[n/3]}]/(n+1);

a(n)=if(n<0,0,polcoeff(serreverse(x^2/(x+x^2+x^4+x^2*O(x^n))),n+1))

Vec(serreverse(x/(1+x+x^3)+O(x^66))/x) /* Joerg Arndt, Aug 19 2012 */

{a(n)=local(A=1); for(i=1,n,A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(x*A)^(2*k)) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 05 2014

a(n) = (Sum_{i=0..floor(n/3)} C(n+1, 1+2i)*C(n-2i, i))/(n+1). - Emeric Deutsch, Nov 03 2002
a(n) = Sum_{k=0..floor(n/3)} C(n,3k)*C(3k,k)/(2k+1). - Paul Barry, Sep 07 2006
D-finite with recurrence: 2*n*(2*n+3)*a(n) + 2*(1-6*n^2)*a(n-1) + 6*(2*n-1)*(n-1)*a(n-2) - 31*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 13 2012
a(n) ~ (2+3*2^(1/3))^(3/2) * (1+3*2^(-2/3))^n/(4*sqrt(6*Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013
G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (x*A(x))^(2*k). - Paul D. Hanna, Sep 05 2014

Name changed by Paul D. Hanna, Nov 15 2012

A307905 Coefficient of x^n in (1 + n*x + x^3)^n.

Original entry on oeis.org

1, 1, 4, 30, 304, 3875, 59631, 1076383, 22309120, 522262245, 13631508400, 392535959156, 12362973152751, 422774554883590, 15600699362473876, 617888566413340503, 26145122799198386944, 1177107512023013681429, 56185125998674634494980, 2834081165961033246374350

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..385

Cf. A116411, A186925, A307903, A307904.

f:= n -> coeff((1+n*x+x^3)^n,x,n):
map(f, [$0..30]); # Robert Israel, Mar 27 2023

Flatten[{1, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27/(4*n^3)], {n, 1, 20}]}] (* Vaclav Kotesovec, May 05 2019 *)

PARI
```
{a(n) = polcoef((1+n*x+x^3)^n, n)}
```

{a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n,3*k)*binomial(3*k,k))}

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

a(n) ~ c * n^n, where c = Sum_{k>=0} 1/(k!*(2*k)!) = HypergeometricPFQ[{}, {1/2, 1}, 1/4] = 1.52106585051363080966025715155941607334728986626976774617... - Vaclav Kotesovec, May 05 2019

A307903 Coefficient of x^n in (1 + nx + nx^3)^n.

Original entry on oeis.org

1, 1, 4, 36, 448, 6875, 124956, 2624293, 62537728, 1667191653, 49158400000, 1588285928306, 55796298391296, 2117279603738494, 86299754734693696, 3760031421065559375, 174374733095888748544, 8575617145497637681301, 445758339115421869936896, 24417549315693295193935516

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Cf. A092366, A116411, A307904, A307905.

Flatten[{1, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27/(4*n^2)], {n, 1, 20}]}] (* Vaclav Kotesovec, May 05 2019 *)

PARI
```
{a(n) = polcoef((1+n*x+n*x^3)^n, n)}
```

{a(n) = sum(k=0, n\3, n^(n-2*k)*binomial(n,3*k)*binomial(3*k,k))}

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

a(n) ~ exp(3*n^(1/3)/2^(2/3)) * n^(n - 1/6) / (2^(2/3)*sqrt(3*Pi)) * (1 - 79/(36 * 2^(1/3) * n^(1/3))). - Vaclav Kotesovec, May 05 2019

A307904 Coefficient of x^n in (1 + x + n*x^3)^n.

Original entry on oeis.org

1, 1, 1, 10, 49, 151, 901, 5881, 28225, 165565, 1158601, 6993196, 44201521, 320103070, 2200745821, 15118248601, 113390231809, 845797019077, 6250243032145, 48718551529210, 384815404148401, 3021055319338813, 24492293678972725, 202303201125303565, 1669594463059152961

Offset: 0

Seiichi Manyama, May 05 2019

nonn

Cf. A116411, A187018, A307903, A307905.

Table[HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2, 1}, -27*n/4], {n, 0, 25}] (* Vaclav Kotesovec, May 05 2019 *)

PARI
```
{a(n) = polcoef((1+x+n*x^3)^n, n)}
```

{a(n) = sum(k=0, n\3, n^k*binomial(n, 3*k)*binomial(3*k, k))}

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

log(a(n)) ~ (n/3 - 1/2)*log(n) + (log(3) - 2*log(2)/3)*n + (2*n)^(2/3)/3 - (2*n)^(1/3)/9. - Vaclav Kotesovec, May 05 2019

A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 48, 120, 309, 787, 2011, 5215, 13652, 35894, 94823, 251889, 672285, 1801185, 4842757, 13064059, 35349463, 95912989, 260896318, 711338596, 1943690464, 5321704006, 14597781706, 40112702176, 110404515703, 304338523999, 840140172621, 2322386563353

Offset: 0

Emely Hanna Li Lobnig, Dec 09 2024

nonn

Cf. A071879, A116411, A378850.

a:= proc(n) option remember; `if`(n<4, [1$3, 3][n+1],
      (2*(8*n^3+3*n^2-25*n-6)*a(n-1)-2*(n-1)*(12*n^2-9*n-10)*
       a(n-2)+(43*n+13)*(n-1)*(n-2)*a(n-3)-31*(n-1)*(n-2)*
        (n-3)*a(n-4))/(2*(2*n+3)*(n+3)*(n-2)))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Dec 09 2024

a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+1, k)/(k+1)) \\ Thomas Scheuerle, Dec 09 2024

a(n) = hypergeom([4/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 3/2, 2], -27/4). - Peter Luschny, Dec 18 2024

More terms from Alois P. Heinz, Dec 09 2024

A378850 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -2.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 73, 190, 505, 1316, 3431, 9065, 24122, 64325, 172082, 462344, 1246685, 3371135, 9140289, 24847422, 67708743, 184906614, 505986933, 1387240401, 3810083424, 10481797131, 28880894706, 79692785251, 220203155689, 609242057143, 1687666776031

Offset: 0

Emely Hanna Li Lobnig, Dec 09 2024

nonn

			For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0).

Cf. A378849, A116411, A071879, A026325.

a := n -> hypergeom([4/3, 5/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 2/3, 5/2, 2], -27/4):
seq(simplify(a(n)), n = 0..30);  # Peter Luschny, Dec 18 2024

a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+3, k+1)/(2*k+3)) \\ Thomas Scheuerle, Dec 09 2024

A370183 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^3) )^n.

Original entry on oeis.org

1, 2, 6, 23, 98, 432, 1929, 8689, 39442, 180248, 828376, 3824757, 17727989, 82438852, 384429751, 1797017598, 8417950626, 39506701508, 185718513144, 874346516454, 4121841403488, 19454625634936, 91924347974883, 434783188981384, 2058320844378109, 9752580801216182

Offset: 0

Seiichi Manyama, Feb 11 2024

nonn

Cf. A116411, A370184.

Cf. A369481.

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^3)) ). See A369481.

A370184 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^3) )^n.

Original entry on oeis.org

1, 3, 15, 87, 539, 3458, 22659, 150594, 1011131, 6841779, 46577430, 318654900, 2188931699, 15087882943, 104301302218, 722840860787, 5020500381131, 34937184351049, 243539967641271, 1700255814753027, 11886457488148674, 83200718154710607, 583026777685802256

Offset: 0

Seiichi Manyama, Feb 11 2024

nonn

Cf. A116411, A370183.

Cf. A369482, A370196.

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^3)) ). See A369482.

A370287 Coefficient of x^n in the expansion of ( (1+x)^3 + x^3 )^n.

Original entry on oeis.org

1, 3, 15, 87, 531, 3333, 21309, 138015, 902547, 5946153, 39406005, 262404585, 1754316045, 11767931451, 79165530375, 533883963567, 3608242091091, 24432635451465, 165721028062605, 1125743155558677, 7657535953619721, 52151890274636463, 355576809975214095

Offset: 0

Seiichi Manyama, Feb 14 2024

nonn

Cf. A116411, A370285.
Cf. A188686, A370286.
Cf. A192132.

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^3 + x^3) ).

A379462 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -3.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 75, 204, 561, 1499, 4001, 10814, 29364, 79704, 216672, 590764, 1614421, 4419049, 12116139, 33277722, 91546143, 252209535, 695803659, 1922166420, 5316714156, 14723570406, 40820144106, 113293243636, 314759548879, 875342190283, 2436582442381

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0). An example of a path with these steps, but not staying on or above y = -3, is for n=6: DDUUUU.

Cf. A071879, A116411, A378849, A378850.

lista(nn) = my(v=vector(nn+5), w); print1(v[4]=1); for(n=1, nn, w=v; for(i=1, n+3, w[i]+=v[i+2]; w[i+1]+=v[i]); v=w; print1(", ", v[4])); \\ Jinyuan Wang, Jan 07 2025

a(n) = Sum_{k=0..floor(n/3)} 2*binomial(n, k*3)*(binomial(3*k+3, k)/(k+2) - binomial(3*k, k-1)/(k+1)). - Thomas Scheuerle, Jan 07 2025

a(n) ~ 23 * (1 + 3/2^(2/3))^(n + 3/2) / (4 * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025

More terms from Jinyuan Wang, Jan 07 2025

cp's OEIS Frontend

A071879 G.f. satisfies: A(x) = 1 + x*A(x) + x^3*A(x)^3.

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A307905 Coefficient of x^n in (1 + n*x + x^3)^n.

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A307903 Coefficient of x^n in (1 + n*x + n*x^3)^n.

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A307904 Coefficient of x^n in (1 + x + n*x^3)^n.

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A378849 a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.

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A378850 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -2.

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A370183 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^3) )^n.

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A370184 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^3) )^n.

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A071879 G.f. satisfies: A(x) = 1 + xA(x) + x^3A(x)^3.

A307903 Coefficient of x^n in (1 + nx + nx^3)^n.