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%I #27 Jun 27 2023 10:28:03
%S 1,1,7,31,175,991,5881,35617,219871,1376095,8710537,55644337,
%T 358198369,2320792657,15120204295,98984058271,650725327231,
%U 4293779332927,28425752310361,188739799967425,1256510215733185,8385127334900305,56078904057164215,375796823748323215
%N Number of n X 3 0..2 arrays with row sums 3 and column sums n.
%C Inverse binomial transform of the Franel numbers (A000172). - _Paul D. Hanna_, Feb 26 2012
%C 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
%C 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
%C 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
%H Andrew Howroyd, <a href="/A172634/b172634.txt">Table of n, a(n) for n = 0..500</a> (terms 1..99 from R. H. Hardin)
%F From _Paul D. Hanna_, Feb 26 2012: (Start)
%F G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
%F Equals the binomial transform of A002898.
%F 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.
%F (End)
%F 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
%F a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - _Vaclav Kotesovec_, Oct 20 2012
%F 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
%e G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
%e 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) +...
%t 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 *)
%o (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
%o (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
%Y Column k=3 of A328747 and A334549.
%Y Cf. A000172, A002898, A006480, A328725.
%K nonn
%O 0,3
%A _R. H. Hardin_, Feb 06 2010
%E a(0)=1 prepended by _Andrew Howroyd_, May 09 2020
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