A127902 Series reversion of x/(1 + x + x^4).

1, 1, 1, 1, 2, 6, 16, 36, 75, 163, 391, 991, 2498, 6150, 15016, 37116, 93481, 238137, 607921, 1550401, 3959335, 10155615, 26182267, 67753907, 175713561, 456422121, 1187771521, 3097869841, 8097629671, 21207212047, 55628797891, 146129168651, 384401493333, 1012608918421

Offset: 1

Author

Paul Barry, Feb 05 2007

Keywords

• easy
• nonn

Comments

Binomial transform of A002293, with three interpolated zeros (series reversion of x/(1+x^4)).

Also the number of rooted labeled trees where each node has 0, 1, or 4 children. - Patrick Devlin, Mar 04 2012

Number of lattice paths from (0,0) to (n-1,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), H=(1,0) and D=(1,-3); a(7) = 16: HHHHHH, uuuDHH, HuuuDH, uHuuDH, uuHuDH, uuuHDH, HHuuuD, HuHuuD, uHHuuD, HuuHuD, uHuHuD, uuHHuD, HuuuHD, uHuuHD, uuHuHD, uuuHHD. - Alois P. Heinz, Apr 14 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Crossrefs

Cf. A240904.

Programs

Magma

[(&+[Binomial(n,4*k)*Binomial(4*k,k)/(3*k+1): k in [0..Floor(n/4)]]): n in [0..30]]; // G. C. Greubel, Apr 30 2018

Mathematica

Rest[CoefficientList[InverseSeries[Series[x/(1+x+x^4),{x,0,20}],x],x]] (* Vaclav Kotesovec, Aug 20 2013 *)

PARI

for(n=0, 30, print1(sum(k=0,floor(n/4), binomial(n,4*k) *binomial(4*k,k)/(3*k+1)), ", ")) \\ G. C. Greubel, Apr 30 2018

Formula

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

Recurrence: 3*(n-1)*(3*n-7)*(3*n+1)*a(n) = 3*(2*n-3)*(18*n^2 - 54*n + 29)*a(n-1) - 3*(n-2)*(54*n^2 - 216*n + 209)*a(n-2) + 54*(n-3)*(n-2)*(2*n-5)*a(n-3) + 229*(n-4)*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Aug 20 2013

a(n) ~ sqrt(4+3^(3/4))*3^(1/4) * (1+4/3*3^(1/4))^n /(12*sqrt(Pi/2) *n^(3/2)). - Vaclav Kotesovec, Aug 20 2013

G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x)^4). - Ilya Gutkovskiy, Jul 01 2020

Extensions

Offset corrected by Vaclav Kotesovec, Aug 20 2013

More terms from Vincenzo Librandi, Apr 15 2014