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A274785 Diagonal of the rational function 1/(1-(wxyz + wxz + wy + xy + z)). 0
1, 1, 25, 121, 2881, 23521, 484681, 5223625, 97949041, 1243490161, 22061635465, 309799010665, 5331441539425, 79799232449665, 1352284119871465, 21095036702450281, 355125946871044561, 5694209222592780625, 95705961654403180201, 1563714140278617173641, 26311422169994777663761 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

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.

S. Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

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

FORMULA

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.

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

MAPLE

with(combinat):

seq(add(binomial(n+2k, 2k)*binomial(n, 2k)*binomial(2k, k)^2, k = 0..floor(n/2)), n = 0..20); # Peter Bala, Jan 27 2018

PROG

(PARI)

my(x='x, y='y, z='z, w='w);

R = 1/(1-(w*x*y*z+w*x*z+w*y+x*y+z));

diag(n, expr, var) = {

my(a = vector(n));

for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));

for (k = 1, n, a[k] = expr;

for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));

return(a);

};

diag(12, R, [x, y, z, w])

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A025283 A076433 A069668 * A214114 A085692 A087399

Adjacent sequences: A274782 A274783 A274784 * A274786 A274787 A274788

KEYWORD

nonn,easy

AUTHOR

Gheorghe Coserea, Jul 13 2016

STATUS

approved

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Last modified November 29 16:39 EST 2022. Contains 358431 sequences. (Running on oeis4.)