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A081798 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k). 10
1, 7, 115, 2371, 54091, 1307377, 32803219, 844910395, 22188235867, 591446519797, 15953338537885, 434479441772845, 11927609772412075, 329653844941016785, 9163407745486783435, 255982736410338609931, 7181987671728091545787 (list; graph; refs; listen; history; text; internal format)
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.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

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

Column k = 3 of A229142. Cf. A001850, A082488, A082459, A229049, A229674, A229675, A229676, A229677.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A183403 A127877 A082487 * A063399 A220181 A328813

Adjacent sequences:  A081795 A081796 A081797 * A081799 A081800 A081801

KEYWORD

easy,nonn

AUTHOR

Emanuele Munarini, Apr 23 2003

STATUS

approved

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Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)