OFFSET
0,2
COMMENTS
a(n) is also a generalization of Delannoy numbers to 3D; i.e. the number of walks from (0,0,0) to (n,n,n) in a 3D square lattice where each step is in the direction of one of (1,0,0), (0,1,0), (0,0,1) and (1,1,1). - Theodore Kolokolnikov, Jul 04 2010
Diagonal of the rational function 1/(1 - x - y - z - x*y*z). - Gheorghe Coserea, Jul 06 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
E. W. Weisstein, in MathWorld: Multinomial Coefficient.
FORMULA
a(n) = w(n,n,n) where w(i,j,k)=w(i-1,j,k)+w(i,j-1,k)+w(i,j,k-1)+w(i-1,j-1,k-1) and where w(0,0,0)=1 and w(i,j,k)=0 if one of i,j,k is strictly negative. - Theodore Kolokolnikov, Jul 04 2010
G.f.: hypergeom([1/3, 2/3],[1],27*x/(1-x)^3)/(1-x). - Mark van Hoeij, Oct 24 2011
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^n / (1-x)^(3*n+1). - Paul D. Hanna, Sep 22 2013
a(n) ~ c*d^n/(Pi*n), where d = (3*(292 + 4*sqrt(5))^(2/3) + 132 + 20*(292 + 4*sqrt(5))^(1/3)) / (2*(292 + 4*sqrt(5))^(1/3)) = 29.900786688498085... is the root of the equation -1 + 3*d - 30*d^2 + d^3 = 0 and c = 1/(2*sqrt(((81 - 27*sqrt(5))/2)^(1/3) + 3*((3 + sqrt(5))/2)^(1/3) - 6)) = 0.8959908650405192232... is the root of the equation -1 - 72*c^2 - 1296*c^4 + 1728*c^6 = 0. - Vaclav Kotesovec, Sep 23 2013, updated Jul 07 2016
From Peter Bala, Jan 13 2016: (Start)
a(n) = Sum_{k = 0..n} multinomial(n + 2*k, k, k, k, n - k). Cf. A001850(n) = Sum_{k = 0..n} multinomial(n + k, k, k, n - k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 42*x^3 + 639*x^4 + 11571*x^5 + ... appears to have integer coefficients. (End)
Conjecture: n^2*(3*n-4)*a(n) -(3*n-2)*(30*n^2-50*n+13)*a(n-1) +(9*n^3-30*n^2+29*n-6)*a(n-2) -(3*n-1)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Apr 15 2016
Conjecture: (n^2)*a(n) +(-28*n^2+24*n-3)*a(n-1) +3*(-19*n^2+78*n-77)*a(n-2) +(5*n-12)*(n-3)*a(n-3) -2*(n-3)^2*a(n-4)=0. - R. J. Mathar, Apr 15 2016
0 = (2*x+1)*(x^3-3*x^2+30*x-1)*x*y'' + (6*x^4-8*x^3+51*x^2+60*x-1)*y' + (x-1)*(2*x^2+2*x-7)*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016
MAPLE
w := proc(i, j, k) option remember; if i=0 and j=0 and k = 0 then 1; elif i<0 or j<0 or k<0 then 0 else w(i-1, j, k)+w(i, j-1, k)+w(i, j, k-1)+w(i-1, j-1, k-1); end: end: for k from 0 to 10 do lprint(w(k, k, k)):end: # Theodore Kolokolnikov, Jul 04 2010
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 51*n^2-45*n+1,
((3*n-2)*(30*n^2-50*n+13)*a(n-1)+(3*n-1)*(n-2)^2*a(n-3)
-(9*n^3-30*n^2+29*n-6)*a(n-2))/(n^2*(3*n-4)))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 22 2013
MATHEMATICA
f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k] Binomial[n + 2k, k], {k, 0, n}]; Array[f, 17, 0] (* Robert G. Wilson v *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1}, 27*x/(1 - x)^3]/(1 - x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 07 2016 *)
PROG
(Maxima) makelist(sum(binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k), k, 0, n), n, 0, 12);
(PARI) {a(n)=polcoeff(sum(m=0, n, (3*m)!/m!^3*x^m/(1-x+x*O(x^n))^(3*m+1)), n)} \\ Paul D. Hanna, Sep 22 2013
(PARI) a(n) = sum(k = 0, n, binomial(n, k) * binomial(n+k, k) * binomial(n+2*k, k)); \\ Michel Marcus, Jan 14 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, Apr 23 2003
STATUS
approved