A104632 1/n times A104631(n), the coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

1, 2, 6, 20, 73, 281, 1125, 4635, 19525, 83710, 364070, 1602327, 7123041, 31937010, 144255802, 655804649, 2998354717, 13777825186, 63596593430, 294743653360, 1371017707245, 6398580086645, 29952930770185, 140604572777250, 661708404611603, 3121439743413256, 14756658303857332

Offset

1,2

Comments

This sequence may be viewed as a higher-order form of the Motzkin numbers, A001006, which are 1/n times the coefficient of x^(n+1) in the expansion of (1+x+x^2)^n. According to Superseeker, this sequence is the INVERT transform of A104184, which is related to Motzkin numbers also. See A104631 for additional comments.

Alternatively, this sequence corresponds to the number of positive walks with n steps {-2,-1,0,1,2} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

Links

Table of n, a(n) for n=1..27.

C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Formula

a(n) = Sum_{i=0..(2*n+1)/5}((-1)^i*binomial(n,i)*binomial(3*n-5*i,n-1))/n. - Vladimir Kruchinin, Apr 06 2017

Conjecture: 2*n*(2*n+1)*(n-1)*a(n) -(n-1)*(19*n^2-19*n+2)*a(n-1) -5*(n-2)*(2*n^2-3*n-1)*a(n-2) +25*n*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 23 2017

Mathematica

f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1]/n, {n, 30}]
a[ n_] := If[ n < 1, 0, Coefficient[ (1 + x + x^2 + x^3 + x^4)^n, x, 2 n + 1] / n]; (* Michael Somos, Dec 01 2016 *)

Prog

(PARI) a(n) = polcoeff((1+x+x^2+x^3+x^4)^n, 2*n+1)/n \\ Michel Marcus, Sep 24 2016
(Maxima)
a(n):=sum((-1)^i*binomial(n, i)*binomial(3*n-5*i, n-1), i, 0, (2*n+1)/5)/n; /* Vladimir Kruchinin, Apr 06 2017 */

Crossrefs

Cf. A005717 (coefficient of x^(n+1) in the expansion of (1+x+x^2)^n).

Sequence in context: A061396 A230823 A192497 * A194956 A150141 A150142

Adjacent sequences: A104629 A104630 A104631 * A104633 A104634 A104635

Keyword

easy,nonn

Author

T. D. Noe, Mar 17 2005

Status

approved