The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274789 Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + wz + xy + xz + y + z)). 1
1, 9, 241, 9129, 402321, 19321689, 981044401, 51794295849, 2814649754641, 156399050208729, 8845463571211521, 507517525088436729, 29468616564702121041, 1728353228376135226329, 102242911938342248555121, 6093340217607472063134249, 365501683327659682186607121, 22049503920365906645420399769 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv:1507.03227 [math-ph], 2015.
Eric Weisstein's World of Mathematics, Legendre Transform
FORMULA
0 = (-x^2+66*x^3+x^4-132*x^5+x^6+66*x^7-x^8)*y''' + (-3*x+303*x^2-396*x^3-594*x^4+405*x^5+291*x^6-6*x^7)*y'' + (-1+224*x-937*x^2+112*x^3+665*x^4+200*x^5-7*x^6)*y' + (9-169*x+254*x^2+30*x^3+5*x^4-x^5)*y, where y is g.f.
From Vaclav Kotesovec, Mar 19 2023: (Start)
Recurrence: (n-2)*n^3*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(34*n^3 - 102*n^2 + 76*n - 17)*a(n-1) - (2*n - 3)*(134*n^4 - 804*n^3 + 1606*n^2 - 1200*n + 291)*a(n-2) + (2*n - 5)*(2*n - 1)*(34*n^3 - 204*n^2 + 382*n - 211)*a(n-3) - (n-3)^3*(n-1)*(2*n - 1)*a(n-4).
a(n) ~ 17^(1/4) * (33 + 8*sqrt(17))^(n + 1/2) / (16 * Pi^(3/2) * n^(3/2)). (End)
From Peter Bala, Jun 26 2023: (Start)
a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)^2 = Sum_{k = 0..n} binomial(n+k,n-k)*binomial(2*k,k)^3, i.e., a(n) is the Legendre transform of A002894. Cf. A243945.
a(n) = hypergeom([1/2, 1/2, -n, n + 1], [1, 1, 1], -16).
O.g.f.: A(x) = Sum_{n >= 0} binomial(2*n,n)^3 * x^n / (1 - x)^(2*n+1). (End)
MAPLE
seq(simplify(hypergeom([1/2, 1/2, -n, n + 1], [1, 1, 1], -16)), n = 0..20); # Peter Bala, Jun 26 2023
MATHEMATICA
a[n_] := SeriesCoefficient[1/(1 - (w x y z + w x y + w x z + w y + w z + x y + x z + y + z)), {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 16 2018 *)
PROG
(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*y+w*z+x*y+x*z+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
Sequence in context: A211052 A085799 A278858 * A263158 A183903 A251670
KEYWORD
nonn,easy
AUTHOR
Gheorghe Coserea, Jul 14 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 18 23:53 EDT 2024. Contains 373490 sequences. (Running on oeis4.)