A245927 -id:A245927 - 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)

A245926 Expansion of g.f. sqrt( (1-x + sqrt(1-14x+x^2)) / (2(1-14*x+x^2)) ).

Original entry on oeis.org

1, 5, 51, 587, 7123, 89055, 1135005, 14660805, 191253843, 2513963567, 33244446601, 441772827105, 5894323986301, 78912561223553, 1059543126891027, 14261959492731387, 192392702881384275, 2600355510685245087, 35206018016510388345, 477377227987055971905

Offset: 0

Paul D. Hanna, Aug 15 2014

Square each term to form a bisection of A245925.

Limit a(n+1)/a(n) = 7 + 4*sqrt(3).

			G.f.: A(x) = 1 + 5*x + 51*x^2 + 587*x^3 + 7123*x^4 + 89055*x^5 +...
where
A(x)^2 = (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)).
Explicitly,
A(x)^2 = 1 + 10*x + 127*x^2 + 1684*x^3 + 22717*x^4 + 309214*x^5 + 4231675*x^6 + 58117672*x^7 + 800173945*x^8 +...+ A245923(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..870

Column k=3 of A337389.

Cf. A245925, A245927, A245923, A243946, A337422.

A245926 := n -> sqrt(add(binomial(4*n-2*k, 2*n-k)*binomial(4*n-k, k)^2, k=0..2*n)); seq(A245926(n), n=0..20); # Peter Luschny, Aug 17 2014

CoefficientList[Series[Sqrt[(1 - x + Sqrt[1 - 14*x + x^2])/(2*(1 - 14*x + x^2))], {x,0,50}], x] (* G. C. Greubel, Jan 29 2017 *)
a[n_] := (-1)^n Hypergeometric2F1[-n, n + 1/2, 1, 4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 16 2018 *)

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

/* From formula for a(n)^2: */
{a(n)=sqrtint(sum(k=0, 2*n, sum(j=0, 4*n-2*k, (-1)^(j+k)*binomial(4*n-k,j+k)^2*binomial(j+k, k)^2)))}
for(n=0, 20, print1(a(n), ", "))

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

a(n)^2 = Sum_{k=0..2*n} Sum_{j=0..4*n-2*k} (-1)^(j+k) * C(4*n-k,j+k)^2 * C(j+k,k)^2.
a(n) ~ (3*sqrt(3)-5) * (7+4*sqrt(3))^(n+1) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 16 2014
a(n)^2 = C(4*n,2*n)*hyper4F3([-2*n,-2*n,-2*n,-2*n+1/2],[1,-4*n,-4*n],4). - Peter Luschny, Aug 17 2014
a(n)^2 = Sum_{k=0..2*n} binomial(4*n-2*k, 2*n-k)*binomial(4*n-k, k)^2. - Peter Luschny, Aug 17 2014
a(n)^2 = Sum_{k=0..2*n} (-1)^k * C(2*k, k)^2 * C(2*n+k, 2*n-k). - Paul D. Hanna, Aug 17 2014
From Peter Bala, Mar 14 2018: (Start)
a(n) = (-1)^n*P(2*n,sqrt(-3)), where P(n,x) denotes the n-th Legendre polynomial. See A008316.
a(n) = (1/C(2*n,n))*Sum_{k = 0..n} (-1)^(n+k)*C(n,k)*C(n+k,k)* C(2*n+2*k,n+k) = Sum_{k = 0..n} (-1)^(n+k)*C(2*k,k)*C(n,k) *C(2*n+2*k,2*n)/C(n+k,n). 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 * u^(n-k), where u = (1 - sqrt(-3))/2 is a primitive sixth root of unity.
a(n) = (-1)^n*Sum_{k = 0..2*n} C(2*n,k)*C(2*n+k,k)*u^(2*k).
(End)
a(n) = (-1)^n*hypergeom([-n, n + 1/2], [1], 4). - Peter Luschny, Mar 16 2018
a(0) = 1, a(1) = 5 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * (28*n^2-42*n+9) * a(n-1) - (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020
From Peter Bala, May 03 2022: (Start)
Conjecture: a(n) = [x^n] ( (1 + x + x^2)*(1 + x)^2/(1 - x)^2 )^n. Equivalently, a(n) = Sum_{k = 0..n} Sum_{i = 0..k} Sum_{j = 0..n-k-i} C(n, k)*C(k, i)*C(2*n, j)*C(3*n-k-i-j-1, n-k-i-j).
If the conjecture is true then the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Calculation suggests that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

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)

A300946 Rectangular array A(n, k) = (-1)^k*hypergeom([-k, k + n/2 - 1/2], [1], 4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 19, 1, 5, 33, 239, 1, 7, 51, 387, 3011, 1, 9, 73, 587, 4737, 38435, 1, 11, 99, 847, 7123, 59523, 496365, 1, 13, 129, 1175, 10321, 89055, 761121, 6470385, 1, 15, 163, 1579, 14499, 129367, 1135005, 9854211, 84975315

Offset: 0

Peter Luschny, Mar 16 2018

			Array starts:
[0] 1,  1,  19,  239,  3011,  38435,  496365,  6470385, ... [A299864]
[1] 1,  3,  33,  387,  4737,  59523,  761121,  9854211, ... [A299507]
[2] 1,  5,  51,  587,  7123,  89055, 1135005, 14660805, ... [A245926]
[3] 1,  7,  73,  847, 10321, 129367, 1651609, 21360031, ... [A084768]
[4] 1,  9,  99, 1175, 14499, 183195, 2351805, 30539241, ... [A245927]
[5] 1, 11, 129, 1579, 19841, 253707, 3284737, 42924203, ...
[6] 1, 13, 163, 2067, 26547, 344535, 4508877, 59402397, ...

Cf. A299864, A299507, A245926, A084768, A245927.

Cf. A300945.

Arow[n_, len_] := Table[(-1)^k Hypergeometric2F1[-k, k + n/2 - 1/2, 1, 4], {k, 0, len}]; Table[Print[Arow[n, 7]], {n, 0, 6}];

A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2(k+4)x+((k-4)x)^2) (1+(k-4)x+sqrt(1-2(k+4)x+((k-4)x)^2)) )).

Original entry on oeis.org

1, 1, 6, 1, 7, 30, 1, 8, 51, 140, 1, 9, 74, 393, 630, 1, 10, 99, 736, 3139, 2772, 1, 11, 126, 1175, 7606, 25653, 12012, 1, 12, 155, 1716, 14499, 80464, 212941, 51480, 1, 13, 186, 2365, 24310, 183195, 864772, 1787607, 218790, 1, 14, 219, 3128, 37555, 352716, 2351805, 9400192, 15134931, 923780

Offset: 0

Seiichi Manyama, Aug 25 2020

			Square array begins:
     1,     1,     1,      1,      1,      1, ...
     6,     7,     8,      9,     10,     11, ...
    30,    51,    74,     99,    126,    155, ...
   140,   393,   736,   1175,   1716,   2365, ...
   630,  3139,  7606,  14499,  24310,  37555, ...
  2772, 25653, 80464, 183195, 352716, 610897, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A002457, A273055, A337370, A245927, A002458, A243947.
Main diagonal gives A337387.
Cf. A337389, A337464.

T[n_, k_] := Sum[If[k == 0, Boole[n == j], k^(n - j)] * Binomial[2*j, j] * Binomial[2*n + 1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 25 2020 *)

{T(n, k) = sum(j=0, n, k^(n-j)*binomial(2*j, j)*binomial(2*n+1, 2*j))}

T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(2*j,j) * binomial(2*n+1,2*j).
T(0,k) = 1, T(1,k) = k+6 and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (4*(k+4)*n^2-2*(k+4)*n-k-2) * T(n-1,k) - (k-4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 29 2020
For fixed k > 0, T(n,k) ~ (2 + sqrt(k))^(2*n + 3/2) / sqrt(8*k*Pi*n). - Vaclav Kotesovec, Aug 31 2020

A245924 Expansion of (1-x - sqrt(1 - 14x + x^2)) / (6x(1 - 14x + x^2)).

Original entry on oeis.org

1, 18, 279, 4132, 59949, 860022, 12252547, 173756232, 2456093529, 34634926810, 487525847535, 6852798238572, 96216461002117, 1349689029354558, 18918661407653979, 265016591806251664, 3710426585319049905, 51924984423522889122, 726369947645489367751, 10157588028419864394420

Offset: 0

Paul D. Hanna, Aug 16 2014

nonn

Self-convolution of A245927.

Limit a(n+1)/a(n) = 7 + 4*sqrt(3).

			G.f.: A(x) = 1 + 18*x + 279*x^2 + 4132*x^3 + 59949*x^4 + 860022*x^5 +...

G. C. Greubel, Table of n, a(n) for n = 0..870

Cf. A245923, A245927.

CoefficientList[Series[(1 - x - Sqrt[1 - 14*x + x^2])/(6*x*(1 - 14*x + x^2)), {x,0,50}], x] (* G. C. Greubel, Feb 14 2017 *)

{a(n)=polcoeff( (1-x - sqrt(1-14*x+x^2 +x^2*O(x^n))) / (6*x*(1-14*x+x^2 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ (26 + 15*sqrt(3)) * (7 + 4*sqrt(3))^n / 24 * (1 - 1/(3^(1/4)*sqrt(Pi*n/2))). - Vaclav Kotesovec, Aug 17 2014

D-finite with recurrence: (n+1)*a(n) +7*(-4*n-1)*a(n-1) +99*(2*n-1)*a(n-2) +7*(-4*n+5)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Jan 23 2020

A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).

Original entry on oeis.org

1, 3, 33, 387, 4737, 59523, 761121, 9854211, 128772609, 1694927619, 22437369633, 298419470979, 3984500221569, 53376363001731, 717044895641121, 9656091923587587, 130310873022310401, 1761872309456567811, 23861153881099854369, 323634591584064809859

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Cf. A299864, A245926, A084768, A245927, A253283.

Cf. A300945, A300946.

seq(simplify( (-1)^n*hypergeom([-n, n], [1], 4)), n = 0..20); # Peter Bala, Apr 18 2024

a[n_] := (-1)^n Hypergeometric2F1[-n, n, 1, 4]; Table[a[n], {n, 0, 19}]

From Vaclav Kotesovec, Jul 05 2018: (Start)
Recurrence: n*(2*n-3)*a(n) = 2*(14*n^2 - 28*n + 11)*a(n-1) - (n-2)*(2*n-1)*a(n-2).
a(n) ~ 2^(-3/2) * 3^(1/4) * (7 + 4*sqrt(3))^n / sqrt(Pi*n). (End)
From Peter Bala, Apr 18 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k-1)*3^k = R(n, 3) for n >= 1, where R(n, x) denotes the n-th row polynomial of A253283.
a(n) = 3*n* hypergeom([1 - n, n + 1], [2], -3) for n >= 1.
a(n) = (1/2)*(LegendreP(n, 7) - LegendreP(n-1, 7)) for n >= 1.
a(n) = [x^n] ( (1 - x)/(1 - 4*x) )^n.
It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
G.f.: (sqrt(x^2 - 14*x + 1) - x + 1)/(2*sqrt(x^2 - 14*x + 1)) = 1 + 3*x + 33*x^2 + 387*x^3 + .... (End)

A299864 a(n) = (-1)^n*hypergeom([-n, n - 1/2], [1], 4).

Original entry on oeis.org

1, 1, 19, 239, 3011, 38435, 496365, 6470385, 84975315, 1122708899, 14906800361, 198740733581, 2658870294349, 35677678567549, 479965685669059, 6471364940381007, 87425255326277907, 1183139999323074963, 16036589185819644633, 217668383345249016045

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..875

Cf. A299507, A245926, A084768, A245927.

Cf. A300945, A300946.

seq((-1)^n*orthopoly[P](n,0,-3/2,-7),n=0..100); # Robert Israel, Mar 21 2018

a[n_] := (-1)^n Hypergeometric2F1[-n, n - 1/2, 1, 4]; Table[a[n], {n, 0, 19}]

From Robert Israel, Mar 21 2018: (Start)
a(n) = JacobiP(n,0,-3/2,-7).
n*(2*n-3)*(4*n-7)*a(n)+(2*n-5)*(n-1)*(4*n-3)*a(n-2)-(4*n-5)*(28*n^2-70*n+39)*a(n-1) = 0. (End)
a(n) ~ sqrt(3) * (1 + sqrt(3))^(4*n - 1)  / (2^(2*n + 1) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 05 2018

A337467 Expansion of sqrt(2 / ( (1-2x+49x^2) * (1-7x+sqrt(1-2x+49*x^2)) )).

Original entry on oeis.org

1, 3, -21, -139, 531, 6489, -9723, -292293, -135117, 12514313, 29905809, -501239553, -2310673379, 18245192679, 140574917259, -562805403867, -7557237645741, 11275709877369, 371974318253601, 201852054629631, -16932135947326551, -42530838930147813, 709138646702505999

Offset: 0

Seiichi Manyama, Aug 28 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A337464.

Cf. A245927, A337422.

a[n_] := Sum[(-3)^(n-k) * Binomial[2*k, k] * Binomial[2*n+1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Apr 29 2021 *)

N=40; x='x+O('x^N); Vec(sqrt(2/((1-2*x+49*x^2)*(1-7*x+sqrt(1-2*x+49*x^2)))))

{a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

a(0) = 1, a(1) = 3 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (4*n^2-2*n+1) * a(n-1) - 49 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020

cp's OEIS Frontend

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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A245926 Expansion of g.f. sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) ).

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

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A300946 Rectangular array A(n, k) = (-1)^k*hypergeom([-k, k + n/2 - 1/2], [1], 4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

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A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2*(k+4)*x+((k-4)*x)^2) * (1+(k-4)*x+sqrt(1-2*(k+4)*x+((k-4)*x)^2)) )).

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A245924 Expansion of (1-x - sqrt(1 - 14*x + x^2)) / (6*x*(1 - 14*x + x^2)).

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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.

A245926 Expansion of g.f. sqrt( (1-x + sqrt(1-14x+x^2)) / (2(1-14*x+x^2)) ).

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

A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2(k+4)x+((k-4)x)^2) (1+(k-4)x+sqrt(1-2(k+4)x+((k-4)x)^2)) )).

A245924 Expansion of (1-x - sqrt(1 - 14x + x^2)) / (6x(1 - 14x + x^2)).

A337467 Expansion of sqrt(2 / ( (1-2x+49x^2) * (1-7x+sqrt(1-2x+49*x^2)) )).