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A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2. 5
1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In general, we have the binomial identity:

if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).

Limit a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).

From Gheorghe Coserea, Jul 05 2016: (Start)

Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).

Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.

(End).

The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - Peter Bala, Mar 02 2017

LINKS

Table of n, a(n) for n=0..17.

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.

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

W. Zudilin, A generating function of the squares of Legendre polynomials, arXiv:1210.2493v2 [math.CA], 2012.

FORMULA

G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)).  - Paul D. Hanna, Aug 30 2014

a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).

a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).

Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014

a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - Vaclav Kotesovec, Aug 18 2014

From Gheorghe Coserea, Jul 05 2016: (Start)

G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).

0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.

(End)

a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - Peter Bala, Mar 02 2017

a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - Peter Luschny, Mar 14 2018

EXAMPLE

G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...

MATHEMATICA

Table[Sum[2^k * Binomial[2*k, k]^2 * Binomial[n + k, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 18 2014 *)

a[n_] := HypergeometricPFQ[{1/2, -n, n + 1}, {1, 1}, -8];

Table[a[n], {n, 0, 17}] (* Peter Luschny, Mar 14 2018 *)

PROG

(PARI) {a(n) = sum(k=0, n, 2^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A243943, A245944, A243007, A001850, A245925, A243945.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A281996 A017306 A210089 * A202836 A052774 A276960

Adjacent sequences:  A243946 A243947 A243948 * A243950 A243951 A243952

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Aug 17 2014

STATUS

approved

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Last modified May 22 00:03 EDT 2018. Contains 304403 sequences. (Running on oeis4.)