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%I #29 Mar 18 2023 15:14:19
%S 1,1,25,121,2881,23521,484681,5223625,97949041,1243490161,22061635465,
%T 309799010665,5331441539425,79799232449665,1352284119871465,
%U 21095036702450281,355125946871044561,5694209222592780625,95705961654403180201,1563714140278617173641,26311422169994777663761
%N Diagonal of the rational function 1/(1-(wxyz + wxz + wy + xy + z)).
%H Seiichi Manyama, <a href="/A274785/b274785.txt">Table of n, a(n) for n = 0..801</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
%H S. Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.
%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%F 0 = (-x^2+2*x^3+257*x^4+508*x^5+257*x^6+2*x^7-x^8)*y''' + (-3*x+15*x^2+1524*x^3+2286*x^4+789*x^5+3*x^6-6*x^7)*y'' + (-1+16*x+1687*x^2+1168*x^3+217*x^4-8*x^5-7*x^6)*y' + (1+183*x-178*x^2-2*x^3-3*x^4-x^5)*y, where y is the g.f.
%F a(n) = Sum_{k = 0..floor(n/2)} C(n + 2*k,2*k)*C(n,2*k)*C(2*k,k)^2 (apply Eger, Theorem 3 to the set of column vectors S = {[0,0,1,0], [1,1,0,0], [0,1,0,1], [1,0,1,1],[1,1,1,1]}). - _Peter Bala_, Jan 27 2018
%F n^3*(n - 2)*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(2*n^3 - 6*n^2 + 4*n - 1)*a(n-1) + (2*n - 3)*(250*n^4 - 1500*n^3 + 3066*n^2 - 2448*n + 629)*a(n-2) + (2*n - 5)*(2*n - 1)*(2*n^3 - 12*n^2 + 22*n - 11)*a(n-3) - (2*n - 1)*(n - 1)*(n - 3)^3*a(n-4). - _Peter Bala_, Mar 17 2023
%F a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * Pi^(3/2) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Mar 17 2023
%p seq(add(binomial(n+2*k, 2*k)*binomial(n, 2*k)*binomial(2*k, k)^2, k = 0..floor(n/2)), n = 0..20); # _Peter Bala_, Jan 27 2018
%t Table[Sum[Binomial[n + 2*k, 2*k]*Binomial[n, 2*k]*Binomial[2*k, k]^2, {k, 0, n/2}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 17 2023 *)
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1-(w*x*y*z+w*x*z+w*y+x*y+z));
%o diag(n, expr, var) = {
%o my(a = vector(n));
%o for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
%o for (k = 1, n, a[k] = expr;
%o for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
%o return(a);
%o };
%o diag(12, R, [x,y,z,w])
%o (PARI) a(n) = sum(k=0, n\2, binomial(n + 2*k,2*k) * binomial(n,2*k) * binomial(2*k,k)^2) \\ _Andrew Howroyd_, Mar 18 2023
%Y Cf. A268545-A268555.
%K nonn,easy
%O 0,3
%A _Gheorghe Coserea_, Jul 13 2016
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