A143003 -id:A143003 - cp's OEIS frontend

A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0

Offset: 0

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

			G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ...

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 = 0..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. A005928, A006077, A113062, A180318, A227454, A227696.

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[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))}

Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.
Expansion of 2*b(q^4) - b(q) = b(q^2)^3 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...].
Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696.
G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.
a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).

A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 4, 2, 1, 3, 6, 4, 0, -1, 4, 6, 4, 2, 2, 2, 3, 4, 2, 0, 1, 6, 8, 2, 0, 3, 6, 0, -2, 0, 6, 6, 4, 4, 2, 4, 3, 4, 0, -2, 0, 6, 8, 2, 2, -1, 6, 4, 2, 1, 4, 6, 4, 2, 0, 6, 0, 0, 0, 0, 4, 6, 8, 2, 1, 2, 12, 4, -2, -2, 2, 6, 0, 2, 2, 2, 0, 8, 4, 0, 3, 3, 8, 2

Offset: 0

Michael Somos, Sep 21 2013

sign

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

Zagier (2009) refers to Case D corresponding to the Apery numbers (A005258).

			G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ...

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 = 0..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. A005258, A078905, A229802.

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 := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* Michael Somos, Jun 10 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ Re[(3 - I) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, real( (3 - I) * [ 0, 1, I, -I, -1][ d%5 + 1])))};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -3, 2, 2, -3][k%5 + 1], 1 + x * O(x^n)), n))};

A = ModularForms( Gamma1(5), 1, prec=20) . basis(); A[0] + 3*A[1]; # Michael Somos, Jun 10 2014

Expansion of f(-q)^2 * (f(-q^5) / f(-q, -q^4))^5 = f(-q^2, -q^3)^2 * (f(-q^5) / f(-q, -q^4))^3 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 5 sequence [ 3, -2, -2, 3, -2, ...].
Moebius transform is period 5 sequence [ 3, 1, -1, -3, 0, ...]. - Michael Somos, Jun 10 2014
G.f. = g(t(q)) where g(), t() are the g.f. for A005258 and A078905.
G.f.: (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)))^5.

A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!Sum_{k = 0..n+1} (-1)^kC(n+1,k)(2n-k)! for n >= 1.

Original entry on oeis.org

-1, 1, 11, 181, 3539, 81901, 2203319, 67741129, 2346167879, 90449857081, 3843107102339, 178468044946621, 8994348275804891, 488964835817842021, 28523735794360301039, 1777328098986754744081, 117817961601577138782479, 8279178465722546926265329

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 Napier's constant: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)* a(n-1)).

Notice the striking parallels with 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)) = 5*[1/(1*3) + 1/(2^2*3*19) + 1/(3^2*19*147) + ...].

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. A005258, A060475, A086764, A143414, 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((-1)^k*binomial(n+1,k)*(2*n-k)!, k = 0..n+1):
seq(a(n), n = 1..19);
# Alternative
a := n -> `if`(n<2, 2*n-1, (2*n)!/(n-1)!*hypergeom([-n-1], [-2*n], -1)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Nov 14 2018

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

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

a(0):= -1, a(n) = 1/(n-1)!*sum {k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.
Apart from the initial term, this sequence is the second superdiagonal of the square array A060475; equivalently, the second subdiagonal of the square array A086764.
Recurrence relation: a(0) = -1, a(1) = 1, (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) = 0, b(1) = 2. Then b(n) = A143414(n) = 1/(n-1)!*sum {k = 0..n-1} C(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) -> e very rapidly.
For example, b(100)/a(100) - e is approximately 1.934 * 10^(-436). The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^n *2*n^2 leads to rapidly converging series for e and 1/e: e = 2 * Sum_{n >= 1} (-1)^n * n^2/(a(n)*a(n-1)) = 2*[1 + 2^2/(1*11) - 3^2/(11*181) + 4^2/(181*3539) - ...]; 1/e = 1/2 - 2*Sum_{n >= 2} (-1)^n * n^2/(b(n)*b(n-1)) = 1/2 - 2*[2^2/(2*30) - 3^2/(30*492) + 4^2/(492*9620) - ...].
Conjectural congruences: for r >= 0 and odd prime p, calculation suggests that a(p^r*(p+1)) + a(p^r) == 0 (mod p^(r+1)).
a(n) = ((2*n)!/(n-1)!)*hypergeom([-n-1], [-2*n], -1) for n >= 2. - Peter Luschny, Nov 14 2018
a(n) ~ 2^(2*n + 1/2) * n^(n+1) / exp(n + 1/2). - Vaclav Kotesovec, Jul 11 2021

A214262 Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.

Original entry on oeis.org

1, -5, 9, -11, 24, -45, 50, -53, 81, -120, 120, -99, 170, -250, 216, -203, 288, -405, 362, -264, 450, -600, 528, -477, 601, -850, 729, -550, 840, -1080, 962, -821, 1080, -1440, 1200, -891, 1370, -1810, 1530, -1272, 1680, -2250, 1850, -1320, 1944, -2640, 2208

Offset: 1

Michael Somos, Jul 09 2012

Zagier (2009) writes "... associated to the weight 3 Eisenstein series g(z) = Sigma b(n)q^n = q - 5q^2 + 9q^3 - 11q^4 + ...".

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - 5*q^2 + 9*q^3 - 11*q^4 + 24*q^5 - 45*q^6 + 50*q^7 - 53*q^8 + 81*q^9 + ...

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.

Indranil Ghosh, Table of n, a(n) for n = 1..2000
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A111661.

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_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 06 2013 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^5 QPochhammer[ q^3] QPochhammer[ q^6]^4 / QPochhammer[ q^2]^4, {q, 0, n}]; (* Michael Somos, Oct 06 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* Michael Somos, Oct 06 2013 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^3 + A) * eta(x^6 + A)^4 / eta(x^2 + A)^4, n))};

{a(n) = my(A, p, e); if( n<0, 0, A = factor( n); prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==3, 9^e, if( p==2, -(4^(e+1) + 9*(-1)^(e+1)) / 5, if( p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1)))))))};

Expansion of (1/9) * c(q) * b(q)^2 * c(q^2) / b(q^2) = (c(q)^3 - 8*c(q^2)^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - Michael Somos, Oct 06 2013
a(n) is multiplicative with a(3^e) = 9^e, a(2^e)  = -(4^(e+1) + 9*(-1)^(e+1)) / 5, a(p^e)  = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e)  = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111661.
G.f.: Sum_{k>0} -(-1)^k * k^2 * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker( -3, k) * (x^k - x^(2*k)) / (1 + x^k)^3.

A260832 a(n) = numerator(Jtilde2(n)).

Original entry on oeis.org

1, 3, 41, 147, 8649, 32307, 487889, 1856307, 454689481, 1748274987, 26989009929, 104482114467, 6488426222001, 25239009088827, 393449178700161, 1535897056631667, 1537112996582116041, 6016831929058214523, 94316599529950360769, 369994845516850143483, 23244865440911268112681

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde2(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(2), 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..830
Takashi Ichinose and 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 and 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 1).
Ling Long, Robert Osburn and Holly Swisher, On a conjecture of Kimoto and Wakayama, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.

Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).

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 -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
seq(a(n), n = 0..20); # Peter Luschny, Dec 08 2022

Numerator[Table[Sum[ (-1)^k*Binomial[-1/2, k]^2*Binomial[n, k], {k, 0, n}], {n,0,50}]] (* G. C. Greubel, Feb 15 2017 *)

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

a(n) = numerator(sum(k=0, n, binomial(2*k, k)*binomial(4*k, 2*k)* binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))) / (2^(4*n)* binomial(2*n,n)));

Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
And 4n^2*J2(n) - (8n^2-8n+3)*J2(n-1) + 4(n-1)^2*J2(n-2) = 0 with J2(0) = 3*zeta(2) and J2(1) = 9*zeta(2)/4.
Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
Jtilde2(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(4*k,2*k)*binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))/(2^(4*n)*binomial(2*n,n)).
From Andrey Zabolotskiy, Oct 04 2016 and Dec 08 2022: (Start)
Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
Jtilde2(n) = A143583(n) / 16^n. (End)
a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - Peter Luschny, Dec 08 2022

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

Original entry on oeis.org

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

cp's OEIS Frontend

A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

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A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

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A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!*Sum_{k = 0..n+1} (-1)^k*C(n+1,k)*(2*n-k)! for n >= 1.

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A214262 Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.

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A260832 a(n) = numerator(Jtilde2(n)).

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

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A143413 Apéry-like numbers for the constant e: a(n) = 1/(n-1)!Sum_{k = 0..n+1} (-1)^kC(n+1,k)(2n-k)! for n >= 1.

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)!.