A212297 -id:A212297 - cp's OEIS frontend

A212296 a(n) = numerator(1 + Sum_{k=1..n} n^2 / Product_{j=1..k} 4*j^2).

5, 33, 869, 48449, 1504375, 124787549, 119224713221, 10003972882859, 9610660732622149, 3811875515172994001, 40741092389313992153, 1050927826884051298685761, 707754011714996709527574437, 184405400463251288725766546203, 496687160874729261988243149308101

Offset: 1

Charles R Greathouse IV, Jul 02 2012

			r(n) = 5/4, 33/16, 869/256, 48449/9216, 1504375/196608, 124787549/11796480, ....
From _Petros Hadjicostas_, Sep 26 2019: (Start)
a(3) = numerator(1 + 3^2/(4*1^2) + 3^2/(4*1^2 * 4*2^2) + 3^2/(4*1^2 * 4*2^2 * 4*3^2)) = numerator(1 + 9/4 + 9/64 + 9/2304) = numerator(869/256) = 869.
a(4) = numerator(1 + 4^2/(4*1^2) + 4^2/(4*1^2 * 4*2^2) + 4^2/(4*1^2 * 4*2^2 * 4*3^2) + 4^2/(4*1^2 * 4*2^2 * 4*3^2 * 4*4^2)) = numerator(1 + 16/4 + 16/64 + 16/2304 + 16/147456) = denominator(48449/9216) = 48449.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..225

Denominators are A212297.

a := n -> numer(1 + add(n^2 / mul(4*j^2, j=1..k), k=1..n)):
seq(a(n), n=1..15); # Peter Luschny, Sep 26 2019

G[n_] := Module[{N=1, D=1}, Sum[N*=2*k-1; D*=2*k; (n/D)^2, {k, 1, n}] + 1]; a[n_] := Numerator[G[n]]; Array[a, 15] (* Jean-François Alcover, Sep 05 2015, translated from PARI *)

G(n)=my(N=1,D=1); sum(k=1,n, N*=2*k-1; D*=2*k; (n/D)^2)+1
a(n)=numerator(G(n))
vector(15, n, a(n))

Redefinition according to the data by Peter Luschny, Sep 26 2019

A213397 Number of (w,x,y) with all terms in {0,...,n} and 2*w >= |x+y-z|.

Original entry on oeis.org

1, 5, 18, 43, 83, 144, 229, 341, 486, 667, 887, 1152, 1465, 1829, 2250, 2731, 3275, 3888, 4573, 5333, 6174, 7099, 8111, 9216, 10417, 11717, 13122, 14635, 16259, 18000, 19861, 21845, 23958, 26203, 28583, 31104, 33769, 36581, 39546, 42667, 45947, 49392, 53005, 56789, 60750

Offset: 0

Clark Kimberling, Jun 12 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A212959, A213396.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w >= Abs[x + y - w], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212297 *)
CoefficientList[Series[(1 + 2 x + 6 x^2 + 2 x^3 + x^4)/((1 - x)^4*(1 + x + x^2)), {x, 0, 44}], x] (* Michael De Vlieger, Dec 22 2017 *)

first(n) = Vec((1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x + x^2)) + O(x^n)) \\ Iain Fox, Dec 22 2017

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: (1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^4*(1 + x + x^2)).
a(n) = (n+1)^3 - A213396(n).
a(n) = floor(2*n^3/3) + 2*n*(n + 1) + 1. - Bruno Berselli, Dec 22 2017

cp's OEIS Frontend

A212296 a(n) = numerator(1 + Sum_{k=1..n} n^2 / Product_{j=1..k} 4*j^2).

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