OFFSET
1,5
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.
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
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x/(1+x+x^4), {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Aug 20 2013 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 05 2007
EXTENSIONS
Offset corrected by Vaclav Kotesovec, Aug 20 2013
More terms from Vincenzo Librandi, Apr 15 2014
STATUS
approved