A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215

Offset: 0

Author

R. H. Hardin, Feb 06 2010

Keywords

• nonn

Comments

Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012

a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019

a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019

a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)

Crossrefs

Column k=3 of A328747 and A334549.

Cf. A000172, A002898, A006480, A328725.

Programs

Mathematica

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

PARI

a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020

Formula

From Paul D. Hanna, Feb 26 2012: (Start)

G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).

Equals the binomial transform of A002898.

a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.

(End)

Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012

G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

Extensions

a(0)=1 prepended by Andrew Howroyd, May 09 2020