A005258 -id:A005258 - cp's OEIS frontend

A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618.

1, 2, 22, 336, 6006, 117348, 2428272, 52303680, 1160427510, 26337699740, 608642155660, 14272471122560, 338764038330480, 8123136091556640, 196484811079765440, 4788469475873867520, 117465323079289162230, 2898183118626011393100

Offset: 1

Lynette O'Brien, Dec 15 2016

nonn

G.f. is the square root of the g.f. for A183204.
This sequence is c_n in Theorem 6.1 in O'Brien's thesis.
Also see Conjecture 5.4 in Chan, Cooper and Sica's paper.

			G.f. = 1 + 2*x + 22*x^2 + 336*x^3 + 6006*x^4 + ....

L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

G. C. Greubel, Table of n, a(n) for n = 1..500
H. H. Chan, S. Cooper, F. Sica, Congruences satisfied by Apéry-like numbers, International Journal of Number Theory, 2010, 6(01), 89-97. Conjecture 5.4.
Lynette O'Brien, Modular forms and two new integer sequences at level 7
Lynette O'Brien, Modular forms and two new integer sequences at level 7

Cf. A183204, A279613, A279618.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

I:=[2, 22]; [1] cat [n le 2 select I[n] else ((26*n^2-39*n+15)* Self(n-1) + 3*(3*n-4)*(3*n-5)*Self(n-2))/n^2 : n in [1..50]] // G. C. Greubel, Jul 04 2018

RecurrenceTable[{a[n+1] == ((26*n^2+13*n+2)*a[n] + 3*(3*n-1)*(3*n-2)*a[n-1])/ (n + 1)^2, a[-1] == 0, a[0] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)
CoefficientList[Series[Sqrt[7]*(1/(25 - 80*x + 24*Sqrt[1 - 27*x]*Sqrt[1+x]))^(1/4) * Hypergeometric2F1[1/12, 5/12, 1, 13824*x^7/(1 - 21*x + 8*x^2 + Sqrt[1 - 27*x] * (1 - 8*x)*Sqrt[1+x])^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 04 2018 *)

(n+1)^2*a_7(n+1) = (26*n^2+13*n+2)*a_7(n) + 3*(3*n-1)*(3*n-2)*a_7(n-1), a(0)=1, a(-1)=0.
Conjecture: For any positive integer n and any prime p with p equiv. 0,1,2 or 4 modulo 7, a(n) equiv. a(n)=a(n_0)a(n_1)...a(n_r) modulo p, where n=n_0+n_1p+...n_rp^r is the base p representation of n.
Conjecture: a(n)~ C n^(-3/2) 27^n where C=0.0955223052681267146513079107870296256727946666510071798669948234917659...

A227454 Expansion of q * (f(q^9) / f(q))^3 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 9, -22, 51, -108, 221, -429, 810, -1476, 2631, -4572, 7802, -13056, 21519, -34918, 55935, -88452, 138332, -213990, 327852, -497592, 748833, -1117692, 1655719, -2434938, 3556791, -5161808, 7445631, -10677096, 15226658, -21599469, 30485268, -42817788

Offset: 1

Michael Somos, Sep 22 2013

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Zagier (2009) denotes the g.f. as t(z) in Case B which is associated with F(t) the g.f. of A006077.

			G.f. = q - 3*q^2 + 9*q^3 - 22*q^4 + 51*q^5 - 108*q^6 + 221*q^7 - 429*q^8 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A006077, A121589, A226535, A227498.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q^9] / QPochhammer[ -q])^3, {q, 0, n}]

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^36 + A)))^3, n))}

Expansion of c(-q^3) / (-3 * b(-q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q) * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9) * eta(q^36)))^3 in powers of q.
Euler transform of period 36 sequence [ -3, 6, -3, 3, -3, 6, -3, 3, 0, 6, -3, 3, -3, 6, -3, 3, -3, 0, -3, 3, -3, 6, -3, 3, -3, 6, 0, 3, -3, 6, -3, 3, -3, 6, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227498.
G.f. t(q) satisfies f(q) = F(t(q)) where F() is the g.f. of A006077 and f() is the g.f. of A226535
G.f.: x * (Product_{k>0} (1 - (-x)^(9*k)) / (1 - (-x)^k))^3.
a(n) = -(-1)^n * A121589(n).

A264541 a(n) = numerator(Jtilde3(n)).

Original entry on oeis.org

0, 1, 65, 13247, 704707, 660278641, 357852111131, 309349386395887, 240498440880062263, 148443546307725010253, 61760947097005048531, 13658972396318235617977, 723464275788899734058353751, 489812222050789870424202126629, 2614176630672654770175367214389, 204702102697072009862200307064701369

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde3(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(3), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..345
Takashi Ichinose, Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).

Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264542 (denominators).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Numerator[Table[-2*Sum[(-1)^k*Binomial[-1/2, k]^2*Binomial[n, k]*Sum[ 1/(Binomial[-1/2, j]^2*(2*j + 1)^3), {j, 0, k - 1}], {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Oct 24 2017 *)

a(n) = numerator(-2*sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)*sum(j=0, k-1, 1/(binomial(-1/2,j)^2*(2*j+1)^3))));

Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).

4n^2*J3(n) - (8n^2-8n+3)*J3(n-1) + 4(n-1)^2*J3(n-2) = 2^n*(n-1)!/(2n-1)!! with J3(0)=7*zeta(3) and J3(1)=21*zeta(3)/4 + 1/2.

A264542 a(n) = denominator(Jtilde3(n)).

Original entry on oeis.org

1, 2, 96, 17280, 860160, 774144000, 408748032000, 347163328512000, 266621436297216000, 163172319013896192000, 67488959156767948800, 14865958099336613068800, 785345441564243189248819200, 530893518497428395932201779200, 2831432098652951444971742822400, 221701133324526098141287462993920000

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde3(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(3), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..345
Takashi Ichinose, Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).

Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264541 (numerators).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Denominator[Table[-2*Sum[(-1)^k*Binomial[-1/2, k]^2*Binomial[n, k]* Sum[1/(Binomial[-1/2, j]^2*(2*j + 1)^3), {j, 0, k - 1}], {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Oct 23 2017 *)

a(n) = denominator(-2*sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)*sum(j=0, k-1, 1/(binomial(-1/2,j)^2*(2*j+1)^3))));

Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).

4n^2*J3(n) - (8n^2-8n+3)*J3(n-1) + 4(n-1)^2*J3(n-2) = 2^n*(n-1)!/(2n-1)!! with J3(0)=7*zeta(3) and J3(1)=21*zeta(3)/4 + 1/2.

A143414 Apéry-like numbers for the constant 1/e: a(n) = (1/(n-1)!)Sum_{k = 0..n-1} binomial(n-1,k)(2*n-k)!.

Original entry on oeis.org

0, 2, 30, 492, 9620, 222630, 5989242, 184139480, 6377545512, 245868202890, 10446648201110, 485126443539012, 24449173476952380, 1329144227959100462, 77535552689576436210, 4831278674685354629040, 320262424087652686405712

Offset: 0

Peter Bala, Aug 14 2008

This sequence satisfies the recursion (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), which leads to a rapidly converging series for the constant 1/e: 1/e = 1/2 - 2 * Sum_{n >= 2} (-1)^n * n^2/(a(n)*a(n-1)).

Notice the striking resemblance to the theory of the Apéry numbers A(n) = A005258(n), which satisfy a similar recurrence relation n^2*A(n) - (n-1)^2*A(n-2) = (11*n^2-11*n+3)*A(n-1) and which appear in the series acceleration formula zeta(2) = 5*Sum_{n>=1} 1/(n^2*A(n)*A(n-1)). Compare with A143413 and A143415.

Seiichi Manyama, Table of n, a(n) for n = 0..365
A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report, Math. Intelligencer 1 (1978/79), no. 4, 195-203.

Cf. A143413, A143415.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a := n -> 1/(n-1)!*add (binomial(n-1,k)*(2*n-k)!,k = 0..n-1): seq(a(n),n = 0..19);
# Alternative:
A143414 := n -> `if`(n=0, 0, ((2*n)!/(n-1)!)*hypergeom([1-n], [-2*n], 1)):
seq(simplify(A143414(n)), n = 0..16); # Peter Luschny, May 14 2020

Table[(1/(n-1)!)*Sum[Binomial[n-1,k]*(2*n-k)!, {k,0,n-1}], {n,0,50}] (* G. C. Greubel, Oct 24 2017 *)

for(n=0,25, print1((1/(n-1)!)*sum(k=0,n-1, binomial(n-1,k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = (1/(n-1)!)*Sum_{k = 0..n-1} binomial(n-1,k)*(2*n-k)!.
Recurrence relation: a(0) = 0, a(1) = 2, (n-1)^2*a(n) - n^2*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1), n >= 2.
Let b(n) denote the solution to this recurrence with initial conditions b(0) = -1, b(1) = 1. Then b(n) = A143413(n) = (1/(n-1)!)*Sum_{k = 0..n+1} (-1)^k*binomial(n+1,k)*(2*n-k)!.
The rational number b(n)/a(n) is equal to the Padé approximation to exp(x) of degree (n+1,n-1) evaluated at x = -1 and b(n)/a(n) -> 1/e very rapidly. For example, |b(100)/a(100) - 1/e| is approximately 2.177 * 10^(-437).
The identity a(n)*b(n-1) - a(n-1)*b(n) = (-1)^n *2*n^2 leads to rapidly converging series for the constants 1/e and e: 1/e = 1/2 - 2*Sum_{n >= 2} (-1)^n * n^2/(a(n)*a(n-1)) = 1/2 - 2*(2^2/(2*30) - 3^2/(30*492) + 4^2/(492*9620) - ...); e = 2 * Sum_{n >= 1} (-1)^n * n^2/(b(n)*b(n-1)) = 2*(1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...).
a(n) = (BesselK(n-1/2,1/2)-(1-2*n)*BesselK(n+1/2,1/2)) * exp(1/2)/(2*Pi^(1/2)). - Mark van Hoeij, Nov 12 2009
a(n) = ((2*n)!/(n-1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - Peter Luschny, May 14 2020
a(n) ~ 2^(2*n + 1/2) * n^(n+1) / exp(n - 1/2). - Vaclav Kotesovec, Jul 11 2021

A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!Sum_{k = 0..n-1} C(n-1,k)(2*n-k)!.

Original entry on oeis.org

0, 1, 5, 41, 481, 7421, 142601, 3288205, 88577021, 2731868921, 94969529101, 3675200329841, 156725471006105, 7302990263511541, 369216917569411601, 20130327811188977621, 1177435382675193700021, 73546210385434763486705

Offset: 0

Peter Bala, Aug 14 2008

This sequence is a modified version of A143414.

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A143413, A143414.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a := n -> 1/(n+1)!*add (binomial(n-1,k)*(2*n-k)!,k = 0..n-1): seq(a(n),n = 0..19);
# Alternative:
A143415 := n -> `if`(n=0, 0, ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)):
seq(simplify(A143415(n)), n = 0..17); # Peter Luschny, May 14 2020

Table[(1/(n+1)!)*Sum[Binomial[n-1,k]*(2*n-k)!, {k,0,n-1}], {n,0,50}] (* G. C. Greubel, Oct 24 2017 *)

for(n=0,25, print1((1/(n+1)!)*sum(k=0,n-1, binomial(n-1,k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!.
a(n) = 1/(n*(n+1))*A143414(n) for n > 0.
Recurrence relation: a(0) = 0, a(1) = 1, (n-1)*(n+1)*a(n) - (n-2)*n*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1) for n >= 2. 1/e = 1/2 - 2 * Sum_{n = 1..inf} (-1)^(n+1)/(n*(n+2)*a(n)*a(n+1)) = 1/2 - 2*[1/(3*1*5) - 1/(8*5*41) + 1/(15*41*481) - 1/(24*481*7421) + ...] .
Conjectural congruences: for r >= 0 and prime p, calculation suggests the congruences a(p^r*(p+1)) == a(p^r) (mod p^(r+1)) may hold.
a(n) = ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - Peter Luschny, May 14 2020

A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

Original entry on oeis.org

2, 6, 54, 564, 6390, 76356, 948276, 12132504, 158984694, 2124923460, 28877309604, 398046897144, 5554209125556, 78328566695736, 1114923122685720, 15999482238880464, 231253045986317814, 3363838379489630916

Offset: 0

Jason Kimberley, Nov 25 2012

This sequence is s_18 in Cooper's paper.
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..830 (terms 0..254 from Jason Kimberley)
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s18 p. 3.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Magma
```
s_18 := func where C is Binomial;
```

Table[Sum[(-1)^j*Binomial[n,j]*Binomial[2j,j]*Binomial[2n-2j, n-j]* (Binomial[2n-3j-1,n] +Binomial[2n-3j,n]), {j,0,Floor[n/3]}], {n,0,20}] (* G. C. Greubel, Oct 24 2017 *)

{a(n) = sum(j=0,floor(n/3), (-1)^j*binomial(n,j)*binomial(2*j,j)* binomial(2*n-2*j,n-j)*(binomial(2*n-3*j-1,n) +binomial(2*n-3*j,n)))}; \\ G. C. Greubel, Apr 02 2019

[sum((-1)^j*binomial(n,j)*binomial(2*j,j)*binomial(2*n-2*j,n-j)* (binomial(2*n-3*j-1,n)+binomial(2*n-3*j,n)) for j in (0..floor(n/3))) for n in (0..20)] # G. C. Greubel, Apr 02 2019

1/Pi
= 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)]
= 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)).
G.f.: 1+hypergeom([1/8, 3/8],[1],256*x^3/(1-12*x)^2)^2/sqrt(1-12*x). - Mark van Hoeij, May 07 2013
Conjecture D-finite with recurrence: n^3*a(n) -2*(2*n-1)*(7*n^2-7*n+3)*a(n-1) +12*(4*n-5)*(n-1)* (4*n-3)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A260667 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)S(k,n)^2, where S(k,x) denotes the polynomial Sum_{j=0..k} binomial(k,j)binomial(x,j)*binomial(x+j,j).

Original entry on oeis.org

1, 37, 1737, 102501, 6979833, 523680739, 42129659113, 3572184623653, 315561396741609, 28807571694394593, 2701627814373536601, 259121323945378645947, 25330657454041707496017, 2516984276442279642274311, 253667099464270541534450025, 25884030861250181046253181349, 2670255662315910532447096232073

Offset: 1

Zhi-Wei Sun, Nov 14 2015

nonn

Conjecture: For k = 0,1,2,... define S(k,x):= Sum_{j=0..k} binomial(k,j)*binomial(x,j)*binomial(x+j,j).
(i) For any integer n > 0, the polynomial (1/n^2) * Sum_{k=0..n-1}(2k+1)*S(k,x)^2 is integer-valued (and hence a(n) is always integral).
(ii) Let r be 0 or 1, and let x be any integer. Then, for any positive integers m and n, we have the congruence
   Sum_{k=0..n-1} (-1)^(k*r)*(2k+1)*S(k,x)^(2m) == 0 (mod n).
(iii) For any odd prime p, we have Sum_{k=0..p-1} S(k,-1/2)^2 == (-1/p)(1-7*p^3*B_{p-3}) (mod p^4), where (a/p) is the Legendre symbol, and B_0,B_1,B_2,... are Bernoulli numbers. Also, for any prime p > 3 we have Sum_{k=0..p-1} S(k,-1/3)^2 == p - (14/3)*(p/3)*p^3*B_{p-2}(1/3) (mod p^4), where B_n(x) denotes the Bernoulli polynomial of degree n; Sum_{k=0..p-1} S(k,-1/4)^2 == (2/p)*p - 26*(-2/p)*p^3*E_{p-3} (mod p^4), where E_0,E_1,E_2,... are Euler numbers; Sum_{k=0..p-1} S(k,-1/6)^2 == (3/p)*p - (155/12)*(-1/p)*p^3*B_{p-2}(1/3) (mod p^4).
Our conjecture is motivated by a conjecture of Kimoto and Wakayama which states that Sum_{k=0..p-1} S(k,-1/2)^2 == (-1/p) (mod p^3) for any odd prime p. The Kimoto-Wakayama conjecture was confirmed by Long, Osburn and Swisher in 2014.
For more related conjectures, see Sun's paper arXiv.1512.00712. - Zhi-Wei Sun, Dec 03 2015

			a(2) = 37 since (1/2^2) * Sum_{k=0..1} (2k+1)*S(k,2)^2 = (S(0,2)^2 + 3*S(1,2)^2)/4 = (1^2 + 3*7^2)/4 = 148/4 = 37.
G.f. = x + 37*x^2 + 1737*x^3 + 102501*x^4 + 6979833*x^5 + 523680739*x^6 + ...

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
K. Kimoto and M. Wakayama, Apéry-like numbers arising from special values of spectral zeta function for non-commutative harmonic oscillators, Kyushu J. Math. 60(2006), no.2, 383-404. (Cf. the formula (6.19).)
L. Long, R. Osburn and H. Swisher, On a conjecture of Kimoto and Wakayama, arXiv:1404.4723 [math.NT], 2014.
Z.-W. Sun, Supercongruences involving products of two binomial coefficients, arXiv:1011.6676 [math.NT], 2010-2013; Finite Fields Appl. 22(2013), 24-44.
Z.-W. Sun, Congruences involving g_n(x)=sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.
Z.-W. Sun, Supercongruences involving dual sequences, arXiv:1502.00712 [math.NT], 2015.

Cf. A000290, A027641, A027642, A122045.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

# Implementing Mark van Hoeij's formula.
c := n -> binomial(2*n, n)/(n + 1):
h := n -> simplify(hypergeom([-n,-n,-n], [1,-2*n], 1)):
b := n -> c(n)^2*((n+11)*(2+4*n)^2*h(n+1)^2-2*(n+1)*(11*n+16)*(1+2*n)*h(n)*h(n+1)-h(n)^2*(n+1)^3)/(25*(n+2)):
a := n -> b(n-1): seq(a(n), n = 1..17);  # Peter Luschny, Nov 11 2022

S[k_,x_]:=S[k,x]=Sum[Binomial[k,j]Binomial[x,j]Binomial[x+j,j],{j,0,k}]
a[n_]:=a[n]=Sum[(2k+1)*S[k,n]^2,{k,0,n-1}]/n^2
Do[Print[n," ",a[n]],{n,1,17}]

a(n) ~ phi^(10*n + 3) / (10 * Pi^2 * n^3), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 06 2021
Conjecture: a(p-1) == 1 (mod p^3) for all primes p >= 5. - Peter Bala, Aug 15 2022
a(n) = ((n+10)*A005258(n)^2 - (11*n+5)*A005258(n)*A005258(n-1) - n*A005258(n-1)^2)/(25*(n+1)). - Mark van Hoeij, Nov 11 2022

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

Original entry on oeis.org

2, 0, 9, 4, 8, 6, 8, 6, 2, 2, 0, 1, 0, 0, 3, 6, 9, 9, 3, 8, 5, 0, 2, 4, 9, 2, 9, 3, 7, 3, 2, 9, 4, 1, 6, 3, 0, 2, 9, 6, 7, 5, 8, 7, 4, 8, 5, 6, 7, 7, 8, 1, 8, 2, 7, 4, 0, 1, 2, 7, 5, 8, 7, 8, 3, 7, 4, 3, 8, 0, 0, 7, 8, 7, 6, 8, 4, 6, 8, 1, 5, 6, 3, 2, 0, 6, 0, 4, 4, 2, 3, 2, 0, 9, 0, 4, 3, 1, 3, 6, 9, 3, 1

Offset: 1

Jean-François Alcover, Sep 14 2015

The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.

			2.09486862201003699385024929373294163029675874856778182740127587837438...

G. C. Greubel, Table of n, a(n) for n = 1..5000
David Bailey, Jonathan Borwein, David Bradley, Experimental determination of Apéry-like identities for zeta(2n+2), arXiv:math/0505270 [math.NT], 2005.

Cf. A013663.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First

zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ Michel Marcus, Sep 14 2015

Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).

A005430 Apéry numbers: nC(2n,n).

Original entry on oeis.org

0, 2, 12, 60, 280, 1260, 5544, 24024, 102960, 437580, 1847560, 7759752, 32449872, 135207800, 561632400, 2326762800, 9617286240, 39671305740, 163352435400, 671560012200, 2756930576400, 11303415363240, 46290177201840, 189368906734800, 773942488394400

Offset: 0

Simon Plouffe

Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006
The aerated sequence 1,0,2,0,12,0,60,0,... has e.g.f. 1+x*Bessel_I(1,2x). - Paul Barry, Mar 29 2010
Conjecture: the terms of the inverse binomial transform are 2*A132894(n). - R. J. Mathar, Oct 21 2012

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Kunle Adegoke, Robert Frontczak, and Taras Goy, Fibonacci-Catalan Series, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A110.
Laurent Alonso and Edward M. Reingold, Analysis of Boyer and Moore's MJRTY Algorithm, 2012.
T. Amdeberhan and Henri Cohen, Bernoulli sum meets golden number, MathOverflow, version of 2017-06-15.
Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Benjamin Ruoyu Kan, Polynomial Approximations for Quantum Hamiltonian Complexity, Bachelor's thesis, Harvard Univ., 2023.
Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, 40(2):175-180, May 2002.
Alfred J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
I. J. Zucker, On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums, J. Number Theory, Vol. 20, No. 1 (1985), 92-102.
Wadim Zudilin, An elementary proof of Apery's theorem, arXiv:math/0202159 [math.NT], 2002.

Cf. A002011, A002457, A002736, A005258, A005259, A005429, 1/beta(n, n+1) in A061928.

Cf. A001803, A003506.

List([0..30], n-> n*Binomial(2*n,n)); # G. C. Greubel, Dec 09 2018

[n*Binomial(2*n,n): n in [0..30]]; // G. C. Greubel, Dec 09 2018

Maple
```
A005430 := n -> n*binomial(2*n, n);
```

Table[n*Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, May 29 2015 *)

a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1,2*x)),2*n)) \\ Ralf Stephan

[n*binomial(2*n,n) for n in range(30)] # G. C. Greubel, Dec 09 2018

a(n) = A002011(n-1)/2 = 2 * A002457(n-1).
Sum_{n >= 1} 1/a(n) = Pi*sqrt(3)/9. - Benoit Cloitre, Apr 07 2002
G.f.:  2*x/sqrt((1-4*x)^3). - Marco A. Cisneros Guevara, Jul 25 2011
E.g.f.: a(n) = n!* [x^n] exp(2*x)*2*x*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
D-finite with recurrence (-n+1)*a(n) + 2*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
G.f.: 2*x*(1-4*x)^(-3/2) = -G(0)/2 where G(k) =  1 - (2*k+1)/(1 - 2*x/(2*x - (k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
a(n-1) = Sum_{k=0..floor(n/2)} k*C(n,k)*C(n-k,k)*2^(n-2*k). - Robert FERREOL, Aug 29 2015
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)/sqrt(5) = A086466, where phi is the golden ratio. (End)
1/a(n) = (-1)^n*Sum_{j=0..n-1} binomial(n-1,j)*Bernoulli(j+n)/(j+n) for n >= 1. See the Amdeberhan & Cohen link. - Peter Luschny, Jun 20 2017
1/a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*HarmonicNumber(n+k) for n >= 1. - Peter Luschny, Aug 15 2017
Sum_{n>=1} x^n/a(n) = 2*sqrt(x/(4-x))*arcsin(sqrt(x)/2), for abs(x) < 4 (Adegoke et al., 2022, section 6, p. 11). - Amiram Eldar, Dec 07 2024

More terms from James Sellers, May 01 2000

cp's OEIS Frontend

A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618.

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A227454 Expansion of q * (f(q^9) / f(q))^3 in powers of q where f() is a Ramanujan theta function.

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A264541 a(n) = numerator(Jtilde3(n)).

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A264542 a(n) = denominator(Jtilde3(n)).

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A143414 Apéry-like numbers for the constant 1/e: a(n) = (1/(n-1)!)*Sum_{k = 0..n-1} binomial(n-1,k)*(2*n-k)!.

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A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!*Sum_{k = 0..n-1} C(n-1,k)*(2*n-k)!.

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A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

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A143414 Apéry-like numbers for the constant 1/e: a(n) = (1/(n-1)!)Sum_{k = 0..n-1} binomial(n-1,k)(2*n-k)!.

A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!Sum_{k = 0..n-1} C(n-1,k)(2*n-k)!.

A260667 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)S(k,n)^2, where S(k,x) denotes the polynomial Sum_{j=0..k} binomial(k,j)binomial(x,j)*binomial(x+j,j).

A005430 Apéry numbers: nC(2n,n).