A243947 -id:A243947 - cp's OEIS frontend

A245927 G.f.: sqrt( (1-x - sqrt(1-14x+x^2)) / (6x(1-14x+x^2)) ).

1, 9, 99, 1175, 14499, 183195, 2351805, 30539241, 400000275, 5274560891, 69929215641, 931226954949, 12446852889901, 166888293332805, 2243683808486451, 30235162687458327, 408274269493595283, 5523024440001832875, 74834275541765522505, 1015429462194625633125

Offset: 0

Paul D. Hanna, Aug 15 2014

Multiply the square of each term by -3 to form a bisection of A245925.

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

			G.f.: A(x) = 1 + 9*x + 99*x^2 + 1175*x^3 + 14499*x^4 + 183195*x^5 +... where
A(x)^2 = (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)). Explicitly,
A(x)^2 = 1 + 18*x + 279*x^2 + 4132*x^3 + 59949*x^4 + 860022*x^5 + 12252547*x^6 + 173756232*x^7 + 2456093529*x^8 +...+ A245924(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..850

Cf. A245925, A245926, A245924, A243947.

seq(add( binomial(2*n+1,2*k)*binomial(2*k,k)*3^(n-k), k = 0..n),n = 0..20); # Peter Bala, Mar 17 2018

CoefficientList[Series[Sqrt[(1-x-Sqrt[1-14x+x^2])/(6x(1-14x+x^2))],{x,0,20}],x] (* Harvey P. Dale, Oct 23 2015 *)
a[n_] := (-1)^n Hypergeometric2F1[-n, n + 3/2, 1, 4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 17 2018 *)

/* From definition: */
{a(n)=polcoeff( sqrt( (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), ", "))

/* From formula for a(n)^2: */
{a(n)=sqrtint((-1/3)*sum(k=0, 2*n+1, sum(j=0, 4*n-2*k+2, (-1)^(j+k)*binomial(4*n-k+2,j+k)^2*binomial(j+k, k)^2)))}
for(n=0, 20, print1(a(n), ", "))

a(n)^2 = (-1/3)*Sum_{k=0..2*n+1} Sum_{j=0..4*n-2*k+2} (-1)^(j+k) * C(4*n-k+2,j+k)^2 * C(j+k,k)^2.
a(n) ~ (3-sqrt(3)) * (7+4*sqrt(3))^(n+1) / (12*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 16 2014
From Peter Bala, Mar 17 2018: (Start)
a(n) = Sum_{k = 0..n} C(2*n+1,2*k)*C(2*k,k)*3^(n-k).
a(n) = 3^n*hypergeom([-n, -n-1/2], [1], 4/3).
n*(4*n-3)*(2*n+1)*a(n) = (4*n-1)*(28*n^2-14*n-5)*a(n-1) - (n-1)*(2*n-1)*(4*n+1)*a(n-2). (End)
a(n) = (-1)^n*hypergeom([-n, n + 3/2], [1], 4). Peter Luschny, Mar 17 2018
a(n) = (-1)^n/sqrt(-3) * P(2*n+1, sqrt(-3)), where P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 17 2024

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

A243945 a(n) = Sum_{k=0..n} C(2k, k)^2 C(n+k, n-k).

Original entry on oeis.org

1, 5, 49, 605, 8281, 120125, 1809025, 27966125, 440790025, 7051890125, 114160867129, 1865975723045, 30743797894681, 509948702030045, 8507207970913729, 142626515754330125, 2401552098016698025, 40591712338241826125, 688413807606268692025, 11710401759994742685125

Offset: 0

Paul D. Hanna, Aug 17 2014

nonn

The g.f.s formed from a(2*n)^(1/2) and (a(2*n+1)/5)^(1/2) are:
A243946: sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) );
A243947: sqrt( (1+x - sqrt(1-18*x+x^2)) / (10*x*(1-18*x+x^2)) ).
Lim_{n->infinity} a(n+1)/a(n) = 9 + 4*sqrt(5).
Diagonal of rational function 1/(1 - (x + y + x*z + y*z + x*y*z)). - Gheorghe Coserea, Aug 24 2018

			G.f.: A(x) = 1 + 5*x + 49*x^2 + 605*x^3 + 8281*x^4 + 120125*x^5 + ... where
A(x) = 1/(1-x) + 2^2*x/(1-x)^3 + 6^2*x^2/(1-x)^5 + 20^2*x^3/(1-x)^7 + 70^2*x^4/(1-x)^9 + 252^2*x^5/(1-x)^11 + 924^2*x^6/(1-x)^13 + ... + A000984(n)^2*x^n/(1-x)^(2*n+1) + ...

Seiichi Manyama, Table of n, a(n) for n = 0..800
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.

Cf. A243946, A243947, A245925.

&cat[ [&+[ Binomial(2*k, k)^2 * Binomial(n+k, n-k): k in [0..n]]]: n in [0..30]]; // Vincenzo Librandi, Aug 25 2018

Table[Sum[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}, -4];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)

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

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

{a(n)=polcoeff( 1 / agm(1-x, sqrt((1-x)^2 - 16*x +x*O(x^n))), n)}
for(n=0,20,print1(a(n),", "))

G.f.: Sum_{n>=0} binomial(2*n, n)^2 * x^n / (1-x)^(2*n+1).
G.f.: 1 / AGM(1-x, sqrt(1-18*x+x^2)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
a(2*n) = A243946(n)^2.
a(2*n+1) = 5 * A243947(n)^2.
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(19*n^2 - 38*n + 14)*a(n-1) - (2*n-3)*(19*n^2 - 38*n + 14)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (2+sqrt(5)) * (9+4*sqrt(5))^n / (4*Pi*n). - Vaclav Kotesovec, Aug 18 2014. Equivalently, a(n) ~ phi^(6*n + 3) / (4*Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -4). - Peter Luschny, Mar 14 2018
G.f. y=A(x) satisfies: 0 = x*(x^2 - 1)*(x^2 - 18*x + 1)*y'' + (3*x^4 - 34*x^3 - 38*x^2 + 38*x - 1)*y' + (x^3 - 3*x^2 - 19*x + 5)*y. - Gheorghe Coserea, Aug 29 2018
From Peter Bala, Feb 07 2022: (Start)
a(n) = P(n,sqrt(5))^2, where P(n,x) denotes the n-th Legendre polynomial.
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. (End)

A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 25, 1, 5, 43, 425, 1, 7, 65, 661, 7025, 1, 9, 91, 965, 10515, 116625, 1, 11, 121, 1345, 15105, 171097, 1951625, 1, 13, 155, 1809, 20995, 243525, 2828101, 32903225, 1, 15, 193, 2365, 28401, 337877, 4001345, 47284251, 558265825

Offset: 0

Peter Luschny, Mar 16 2018

			[0] 1,  1,  25,  425,  7025, 116625,  1951625,  32903225, ... [A299845]
[1] 1,  3,  43,  661, 10515, 171097,  2828101,  47284251, ... [A299506]
[2] 1,  5,  65,  965, 15105, 243525,  4001345,  66622085, ...
[3] 1,  7,  91, 1345, 20995, 337877,  5544709,  92234527, ... [A243946]
[4] 1,  9, 121, 1809, 28401, 458649,  7544041, 125700129, ... [A084769]
[5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947]
[6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...

Cf. A299845, A299506, A243946, A084769, A243947.

Cf. A300946.

Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}];
Table[Print[Arow[n, 7]], {n, 0, 6}];
T[n_, k_] := If[k==0, 1, 4^k*Sum[(5/4)^j*Binomial[k, j]*Binomial[k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)] ,{j, 0, n}]]; Flatten[Table[T[n, k],{n, 0, 8}, {k, 0, n}]] (* Detlef Meya, May 28 2024 *)

T(n, k) = if k = 0 then 1, otherwise 4^k*Sum_{j=0..n} (5/4)^j * binomial(k, j) * binomial(k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)). - Detlef Meya, May 28 2024

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

A299845 a(n) = hypergeom([-n, n - 1], [1], -4).

Original entry on oeis.org

1, 1, 25, 425, 7025, 116625, 1951625, 32903225, 558265825, 9522632225, 163160773625, 2806202183625, 48420275891025, 837813745045425, 14531896733426025, 252593595973313625, 4398859688478578625, 76733590756134492225, 1340547988367851940825, 23451231922182584693225

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..798

Cf. A299506, A243946, A084769, A243947.

Cf. A300945, A300946.

f:= gfun:-rectoproc({4*n*(n-2)^2*a(n)+4*(n-1)^2*(n-3)*a(n-2)-4*(2*n-3)*(9*n^2-27*n+17)*a(n-1)=0,
a(0)=1,a(1)=1,a(2)=25},a(n),remember):
map(f, [$0..100]); # Robert Israel, Mar 21 2018

a[n_] := Hypergeometric2F1[-n, n - 1, 1, -4]; Table[a[n], {n, 0, 19}]
a[0]:=1; a[1]:=1; a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, May 22 2024 *)

4*n*(n-2)^2*a(n) + 4*(n-1)^2*(n-3)*a(n-2) - 4*(2*n-3)*(9*n^2-27*n+17)*a(n-1) = 0.  - Robert Israel, Mar 21 2018
a(n) ~ 2^(-3/2) * 5^(3/4) * phi^(6*n - 3) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 05 2018
a(n) = 4^n*Sum_{k=0..n} (5/4)^k*Gamma(n + 1)*Gamma(n - 1)/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1)) for n >= 2. - Detlef Meya, May 22 2024

A299506 a(n) = hypergeom([-n, n - 1/2], [1], -4).

Original entry on oeis.org

1, 3, 43, 661, 10515, 171097, 2828101, 47284251, 797456947, 13540982665, 231188344401, 3964874384863, 68252711769373, 1178662654873191, 20409993947488075, 354260920943874245, 6161735337225790035, 107368528677807960185, 1873946997372948997345, 32754419073618998202975

Offset: 0

Peter Luschny, Mar 16 2018

nonn

Cf. A299845, A243946, A084769, A243947.

Cf. A300945, A300946.

a[n_] := Hypergeometric2F1[-n, n - 1/2, 1, -4]; Table[a[n], {n, 0, 19}]
a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1/2])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1/2]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* Detlef Meya, May 22 2024 *)

From Vaclav Kotesovec, Jul 05 2018: (Start)
Recurrence: n*(2*n - 3)*(4*n - 7)*a(n) = 9*(4*n - 5)*(4*n^2 - 10*n + 5)*a(n-1) - (n-1)*(2*n - 5)*(4*n - 3)*a(n-2).
a(n) ~ 2^(-3/2) * sqrt(5) * phi^(6*n - 3/2) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. (End)
a(n) = 4^n*Sum_{k=0..n} (5/4)^k*(Gamma(n + 1)*Gamma(n - 1/2))/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1/2)). - Detlef Meya, May 22 2024

cp's OEIS Frontend

A245927 G.f.: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A243945 a(n) = Sum_{k=0..n} C(2*k, k)^2 * C(n+k, n-k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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)) )).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A299845 a(n) = hypergeom([-n, n - 1], [1], -4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A299506 a(n) = hypergeom([-n, n - 1/2], [1], -4).

Views

Author

Keywords

A245927 G.f.: sqrt( (1-x - sqrt(1-14x+x^2)) / (6x(1-14x+x^2)) ).

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

A243945 a(n) = Sum_{k=0..n} C(2k, k)^2 C(n+k, n-k).

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)) )).