A243945 -id:A243945 - cp's OEIS frontend

A245925 G.f.: Sum_{n>=0} x^nSum_{k=0..n} (-1)^k C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j.

1, -3, 25, -243, 2601, -29403, 344569, -4141875, 50737129, -630663003, 7930793025, -100681224075, 1288236350025, -16592960274075, 214939203248025, -2797935722568243, 36578032462268649, -480000660000226875, 6320012816203363489, -83462977778600141643, 1105193229806740453201

Offset: 0

Paul D. Hanna, Aug 15 2014

The g.f.s formed from a(2*n)^(1/2) and (-a(2*n+1)/3)^(1/2) are:
A245926: sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) );
A245927: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).
Lim_{n->infinity} a(n+1)/a(n) = -(7 + 4*sqrt(3)).

			G.f.: A(x) = 1 - 3*x^2 + 25*x^4 - 243*x^6 + 2601*x^8 - 29403*x^10 + ...
where the g.f. is given by the binomial series:
A(x) = 1 + x*(1 - (1+x)) + x^2*(1 - 2^2*(1+x) + (1+2^2*x+x^2))
+ x^3*(1 - 3^2*(1+x) + 3^2*(1+2^2*x+x^2) - (1+3^2*x+3^2*x^2+x^3))
+ x^4*(1 - 4^2*(1+x) + 6^2*(1+2^2*x+x^2) - 4^2*(1+3^2*x+3^2*x^2+x^3) + (1+4^2*x+6^2*x^2+4^2*x^3+x^4))
+ x^5*(1 - 5^2*(1+x) + 10^2*(1+2^2*x+x^2) - 10^2*(1+3^2*x+3^2*x^2+x^3) + 5^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) - (1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5))
+ x^6*(1 - 6^2*(1+x) + 15^2*(1+2^2*x+x^2) - 20^2*(1+3^2*x+3^2*x^2+x^3) + 15^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) - 6^2*(1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5) + (1+6^2*x+15^2*x^2+20^2*x^3+15^2*x^4+6^2*x^5+x^6)) + ...
in which the coefficients of odd powers of x vanish.
We can also express the g.f. by the binomial series identity:
A(x) = 1/(1+x) + x/(1+x)^3*(1-x)^2 + x^2/(1+x)^5*(1 - 2^2*x + x^2)^2
+ x^3/(1+x)^7*(1 - 3^2*x + 3^2*x^2 - x^3)^2
+ x^4/(1+x)^9*(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)^2
+ x^5/(1+x)^11*(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)^2
+ x^6/(1+x)^13*(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)^2 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

Cf. A245926, A245927, A245929, A243945, A249891.

A245925 := n -> (-1)^n*add(binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2, k=0..n); seq(A245925(n), n=0..20); # Peter Luschny, Aug 17 2014

Table[Sum[Sum[(-1)^(j+k) * Binomial[2*n - k, j + k]^2 * Binomial[j + k, k]^2, {j, 0, 2*n - 2*k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 16 2014 after Paul D. Hanna *)
a[n_] := (-1)^n*HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1/2, 1, 1}, 1/4];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Mar 14 2018 *)

/* By definition: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, (-1)^k*binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*x^j)+x*O(x^n))), n)}
for(n=0, 20, print1(a(2*n), ", "))

/* From alternate g.f.: */
{a(n)=local(A=1);A=sum(m=0,n,x^m/(1+x)^(2*m+1)*sum(k=0,m,binomial(m,k)^2*(-x)^k)^2+x*O(x^n));polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))

/* From formula for a(n); printing only nonzero terms: */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, (-1)^(j+k)*binomial(n-k, j+k)^2*binomial(j+k, k)^2))}
for(n=0, 20, print1(a(2*n), ", "))

/* From formula for a(n) (nonzero terms): */
{a(n)=sum(k=0, n, sum(j=0, 2*n-2*k, (-1)^(j+k)*binomial(2*n-k,j+k)^2*binomial(j+k, k)^2))}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n), after Peter Luschny and Robert Israel: */
{a(n) = (-1)^n * sum(k=0,n, binomial(2*k, k) * binomial(n+k, n-k)^2)}
for(n=0, 20, print1(a(n), ", "))

/* Simpler formula for a(n): */
{a(n) = sum(k=0, n, (-1)^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
for(n=0, 20, print1(a(n), ", "))

/* Using AGM: */
{a(n)=polcoeff( 1 / agm(1-x, sqrt(1+14*x+x^2 +x*O(x^n))), n)}
for(n=0,20,print1(a(n),", "))

A245925 = lambda n: (-1)^n*sum(binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2 for k in (0..n))
[A245925(n) for n in range(21)] # Peter Luschny, Aug 17 2014

G.f.: Sum_{n>=0} x^n / (1+x)^(2*n+1) * ( Sum_{k=0..n} C(n,k)^2*(-x)^k )^2.
G.f.: 1 / AGM(1-x^2, sqrt(1+14*x^2+x^4)), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
a(2*n) = A245926(n)^2.
a(2*n+1) = (-3)*A245927(n)^2.
a(n) = Sum_{k=0..n} Sum_{j=0..2*n-2*k} (-1)^(j+k) * C(2*n-k,j+k)^2 * C(j+k,k)^2.
D-finite with recurrence: n^2*(2*n-3)*a(n) = -(2*n-1)*(13*n^2 - 26*n + 10)*a(n-1) + (2*n-3)*(13*n^2 - 26*n + 10)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 16 2014
a(n) ~ (-1)^n * (2+sqrt(3)) * (7+4*sqrt(3))^n / (4*Pi*n). - Vaclav Kotesovec, Aug 16 2014
a(n) = (-1)^n*Sum_{k=0..n} binomial(2*(n-k), n-k)*binomial(2*n-k, k)^2. - Peter Luschny, Aug 17 2014
a(n) = (-1)^n*binomial(2*n,n)*hyper4F3([-n,-n,-n,-n+1/2],[1,-2*n,-2*n], 4). - Peter Luschny, Aug 17 2014
a(n) = Sum_{k=0..n} (-1)^k * C(2*k, k)^2 * C(n+k, n-k). - Paul D. Hanna, Aug 17 2014
a(n) = (-1)^n*hypergeom([-n, -n, n + 1, n + 1], [1/2, 1, 1], 1/4). - Peter Luschny, Mar 14 2018
a(n) = Legendre_P(n, sqrt(-3))^2. - Peter Bala, Dec 22 2020
G.f.: Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2*x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022
From Peter Bala, Apr 05 2022: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n-k,k)*binomial(2*n-2*k,n-k)^2.
a(n) = (-1)^n * Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)^2.
a(n) = (-1)^n*binomial(2*n,n)^2*hypergeom([-n,-n,-n,], [-2*n,-n+1/2], 1/4). (End)

A243946 Expansion of sqrt( (1+x + sqrt(1-18x+x^2)) / (2(1-18*x+x^2)) ).

Original entry on oeis.org

1, 7, 91, 1345, 20995, 337877, 5544709, 92234527, 1549694195, 26237641045, 446925926881, 7650344197987, 131489964887341, 2267722252458475, 39224201631222475, 680160975405238145, 11820134678459908115, 205812328555924135045, 3589742656727603141425, 62707329988264214752675

Offset: 0

Paul D. Hanna, Aug 17 2014

Square each term to form a bisection of A243945.

Limit_{n->oo} a(n+1)/a(n) = 9 + 4*sqrt(5).

			G.f.: A(x) = 1 + 7*x + 91*x^2 + 1345*x^3 + 20995*x^4 + 337877*x^5 + ...,
where A(x)^2 = (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)).

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A243945, A243947, A084769, A245926, A008316.

seq(add(binomial(n,k)*binomial(n+k,k)*binomial(2*n+2*k,n+k), k = 0..n)/binomial(2*n,n), n = 0..20); # Peter Bala, Mar 14 2018

a[n_] := Hypergeometric2F1[-n, n + 1/2, 1, -4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 16 2018 *)
CoefficientList[Series[Sqrt[(1+x+Sqrt[1-18x+x^2])/(2(1-18x+x^2))],{x,0,20}],x] (* Harvey P. Dale, Dec 26 2019 *)
a[n_] := Sum[(5^k Gamma[2 n + 1])/(Gamma[2 k + 1]*Gamma[n - k + 1]^2), {k, 0, n}];
Flatten[Table[a[n], {n, 0, 19}]] (* Detlef Meya, May 22 2024 *)

/* From definition: */
{a(n)=polcoeff( sqrt( (1+x + sqrt(1-18*x+x^2 +x*O(x^n))) / (2*(1-18*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))

/* From a(n) = sqrt( A243945(2*n) ): */
{a(n)=sqrtint( sum(k=0, 2*n, binomial(2*k, k)^2*binomial(2*n+k, 2*n-k)) )}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 5^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))} \\ Seiichi Manyama, Aug 25 2020

from math import comb
def A243946(n): return sum(5**(n-k)*comb(m:=k<<1,k)*comb(n<<1,m) for k in range(n+1)) # Chai Wah Wu, Mar 23 2023

a(n)^2 = Sum_{k=0..2*n} C(2*k, k)^2 * C(2*n+k, 2*n-k).
a(n) ~ sqrt(2+sqrt(5)) * (9+4*sqrt(5))^n / (2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 3/2) / (2*sqrt(2*Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
From Peter Bala, Mar 14 2018: (Start)
a(n) = P(2*n,sqrt(5)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.
a(n) = (1/C(2*n,n))*Sum_{k = 0..n} C(n,k)*C(n+k,k)* C(2*n+2*k,n+k). In general, P(2*n,sqrt(1 + 4*x)) = (1/C(2*n,n))*Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*x^k.
a(n) = Sum_{k = 0..2*n} C(2*n,k)^2 * phi^(2*n-2*k), where phi = (sqrt(5) + 1)/2.
a(n) = Sum_{k = 0..2*n} C(2*n,k)*C(2*n+k,k)*Phi^k, where Phi = (sqrt(5) - 1)/2. (End)
a(n) = hypergeom([-n, n + 1/2], [1], -4). - Peter Luschny, Mar 16 2018
D-finite with recurrence: n*(2*n-1)*(4*n-5)*a(n) -(4*n-3)*(36*n^2-54*n+11)*a(n-1) +(n-1)*(4*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
a(n) = Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*n,2*k). - Seiichi Manyama, Aug 25 2020

A243947 Expansion of g.f. sqrt( (1+x - sqrt(1-18x+x^2)) / (10x(1-18x+x^2)) ).

Original entry on oeis.org

1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, 2849270515, 48395044705, 826479148001, 14177519463191, 244109912494525, 4216385987238575, 73024851218517275, 1267712063327871245, 22052786911315216595, 384321597582115655825, 6708530714274563938225

Offset: 0

Paul D. Hanna, Aug 17 2014

Multiply the square of each term by 5 to form a bisection of A243945.

Limit_{n->oo} a(n+1)/a(n) = 9 + 4*sqrt(5).

			G.f.: A(x) = 1 + 11*x + 155*x^2 + 2365*x^3 + 37555*x^4 + 610897*x^5 + ...
where
A(x)^2 = (1+x - sqrt(1-18*x+x^2)) / (10*x*(1-18*x+x^2)).

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A243945, A243946, A084769, A245927, A008316.

seq(add(1/2*binomial(2*k+1,k)*binomial(n,k)*binomial(2*n+2*k+2,2*n+1)/binomial(n+k+1,n), k = 0..n), n = 0..20); # Peter Bala, Mar 15 2018

CoefficientList[Series[Sqrt[((1+x-Sqrt[1-18x+x^2]))/(10x(1-18x+x^2))],{x,0,20}],x] (* Harvey P. Dale, Jul 31 2016 *)
a[n_] := Hypergeometric2F1[-n, n + 3/2, 1, -4];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 16 2018 *)

/* From definition: */
{a(n)=polcoeff( sqrt( (1+x - sqrt(1-18*x+x^2 +x^2*O(x^n))) / (10*x*(1-18*x+x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))

/* From a(n) = sqrt( A243945(2*n+1)/5 ): */
{a(n)=sqrtint( (1/5)*sum(k=0, 2*n+1, binomial(2*k, k)^2*binomial(2*n+k+1, 2*n-k+1)) )}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A243947(n): return sum(5**(n-k)*comb(m:=k<<1,k)*comb((n<<1)+1,m) for k in range(n+1)) # Chai Wah Wu, Mar 23 2023

a(n)^2 = (1/5) * Sum_{k=0..2*n+1} C(2*k, k)^2 * C(2*n+k+1, 2*n-k+1).
a(n) ~ (9+4*sqrt(5))^(n+1) / (2*5^(1/4)*sqrt(2*Pi*n) * sqrt(5+2*sqrt(5))). - Vaclav Kotesovec, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 9/2) / (2^(3/2) * sqrt(5*Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
From Peter Bala, Mar 15 2018: (Start)
a(n) = (1/sqrt(5))*P(2*n+1,sqrt(5)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.
a(n) = Sum_{k = 0..n} (1/2)*C(2*k+1,k)*C(n,k)*C(2*n+2*k+2,2*n+1)/C(n+k+1,n). In general, (1/sqrt(1 + 4*x))*P(2*n+1,sqrt(1+4*x)) = (1/(2*C(2*n+1,n))) * Sum_{k = 0..n} C(n,k)*C(n+k+1,k)*C(2*n+2*k+2,n+k+1)*x^k.
a(n) = (1/sqrt(5))*Sum_{k = 0..2*n+1} C(2*n+1,k)^2 * phi^(2*n-2*k+1), where phi = (sqrt(5) + 1)/2.
a(n) = (1/sqrt(5))*Sum_{k = 0..2*n+1} C(2*n+1,k)*C(2*n+1+k,k) * Phi^k, where Phi = (sqrt(5) - 1)/2. (End)
a(n) = hypergeom([-n, n + 3/2], [1], -4). - Peter Luschny, Mar 16 2018
From Peter Bala, Mar 17 2018: (Start)
a(n) = Sum_{k = 0..n} C(2*n+1,2*k)*C(2*k,k)*5^(n-k).
D-finite with recurrence: n*(4*n-3)*(2*n+1)*a(n) = (4*n-1)*(36*n^2-18*n-7)*a(n-1) - (n-1)*(2*n-1)*(4*n+1)*a(n-2). (End)

A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

Original entry on oeis.org

1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561

Offset: 0

Paul D. Hanna, Aug 17 2014

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_{n -> oo} 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

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

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, and 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.

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
Related to diagonal of rational functions: A268545 - A268555.
Cf. A001850, A243945, A245925.

[Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023

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 *)
LegendreP[Range[0, 30], 3]^2 (* G. C. Greubel, May 17 2023 *)

{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), ", "))

{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), ", "))

from math import comb
def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # Chai Wah Wu, Mar 23 2023

[gen_legendre_P(n,0,3)^2 for n in range(41)] # G. C. Greubel, May 17 2023

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
G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022

A274789 Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + wz + xy + xz + y + z)).

Original entry on oeis.org

1, 9, 241, 9129, 402321, 19321689, 981044401, 51794295849, 2814649754641, 156399050208729, 8845463571211521, 507517525088436729, 29468616564702121041, 1728353228376135226329, 102242911938342248555121, 6093340217607472063134249, 365501683327659682186607121, 22049503920365906645420399769

Offset: 0

Gheorghe Coserea, Jul 14 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..548
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.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
Eric Weisstein's World of Mathematics, Legendre Transform

Cf. A002894, A243945, A268545-A268555.

seq(simplify(hypergeom([1/2, 1/2, -n, n + 1], [1, 1, 1], -16)), n = 0..20); # Peter Bala, Jun 26 2023

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 *)

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])

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)

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

Original entry on oeis.org

1, 3, 19, 173, 1881, 22655, 291775, 3940725, 55149025, 793387235, 11668476579, 174735112997, 2656296912361, 40897718776647, 636588467802679, 10002872642155085, 158483629611962025, 2529389028336106475, 40631849127696017275, 656509442594976984405, 10663184061320964941761

Offset: 0

Ilya Gutkovskiy, Nov 29 2021

Cf. A000108, A086618, A086621, A243945.

A349768 := proc(n)
    hypergeom([1/2,-n,n+1],[1,2],-4) ;
    simplify(%) ;
end proc:
seq(A349768(n),n=0..20) ; # R. J. Mathar, Mar 02 2023

Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[2 k, k]/(k + 1), {k, 0, n}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2, -n, n + 1}, {1, 2}, -4], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)/(k+1)); \\ Michel Marcus, Nov 29 2021

From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)*(2*n - 3)*a(n) = (2*n - 1)*(19*n^2 - 37*n + 12)*a(n-1) - (2*n - 3)*(19*n^2 - 39*n + 14)*a(n-2) + (n-3)*(n-2)*(2*n - 1)*a(n-3).
a(n) ~ sqrt(5) * phi^(6*n + 3) / (8*Pi*n^2), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence n*(n+1)*a(n) +(n+1)*(n-4)*a(n-1) +2*(-171*n^2 +512*n -388)*a(n-2) +2*(9*n^2 +296*n -796)*a(n-3) +(341*n^2 -2425*n +4320)*a(n-4) -19*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

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A245925 G.f.: Sum_{n>=0} x^n*Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j.

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A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

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A274789 Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + wz + xy + xz + y + z)).

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A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

A245925 G.f.: Sum_{n>=0} x^nSum_{k=0..n} (-1)^k C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j.

A243946 Expansion of sqrt( (1+x + sqrt(1-18x+x^2)) / (2(1-18*x+x^2)) ).

A243947 Expansion of g.f. sqrt( (1+x - sqrt(1-18x+x^2)) / (10x(1-18x+x^2)) ).