A036917 -id:A036917 - cp's OEIS frontend

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)!.

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).

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

Original entry on oeis.org

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821

Offset: 0

N. J. A. Sloane

nonn

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006256.
Cf. A001764, A006013, A007318, A036917, A062236.
Cf. A000302, A308523, A386367, A386368.

a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012

Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - Seiichi Manyama, Jul 19 2025
G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025

A036916 a(n) = Sum_{k=0..n} binomial(2n-2k,n-k)^2 * binomial(n,k)^2.

Original entry on oeis.org

1, 5, 53, 761, 12661, 229705, 4410665, 88127485, 1813270645, 38158684745, 817458330553, 17767242718285, 390819348043369, 8683822363169933, 194618212789162733, 4394243766346694161, 99862206804817230965, 2282427331053360624713

Offset: 0

N. J. A. Sloane

nonn

Cf. M. Petkovsek et al., A=B, Peters, p. ix.

Reinhard Zumkeller, Table of n, a(n) for n = 0..500

Cf. A036915, A036917.
Cf. A007318.
Row n=4 of A275784.

a036916 n = sum $ map
   (\k -> (a007318 (2*n-2*k) (n-k))^2 * (a007318 n k)^2) [0..n]
-- Reinhard Zumkeller, May 24 2012

Table[Sum[Binomial[2n-2k,n-k]^2 Binomial[n,k]^2,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 31 2013 *)

(n - 1)*(144*n^3 - 864*n^2 + 1693*n - 1075)*n^3*a(n) - 2*(n - 1)*(2592*n^6 - 19440*n^5 + 56322*n^4 - 80296*n^3 + 60004*n^2 - 23017*n + 3580)*a(n - 1) + (42336*n^7 - 423360*n^6 + 1769838*n^5 - 4006912*n^4 + 5293968*n^3 - 4062414*n^2 + 1661406*n - 274520)*a(n - 2) - 2*(34848*n^5 - 261360*n^4 + 741842*n^3 - 984642*n^2 + 598948*n - 127215)*(n - 2)^2*a(n - 3) + 225*(144*n^3 - 432*n^2 + 397*n - 102)*(n - 2)^2*(n - 3)^2*a(n - 4) = 0 - Vladeta Jovovic, Jul 15 2004

a(n) ~ 5^(2*n+2) / (16 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 02 2014

A036915 a(n) = Sum_{k=0..n} C(2n-2k,n-k)^2 * C(2*n,k)^2.

Original entry on oeis.org

1, 8, 136, 2996, 76168, 2121808, 62886484, 1947164948, 62250464648, 2038917457664, 68056394358736, 2306354482651568, 79138758509239636, 2743929765899780336, 95983445574557632948, 3383168407446955236196, 120039826640626408744328

Offset: 0

N. J. A. Sloane

nonn

There's an erroneous assertion (caused by a typographical error) in the first printing of M. Petkovsek et al., "A=B", Peters, p. ix. that this is the same as A036917; it is not.

Table[Sum[Binomial[2n-2k,n-k]^2 Binomial[2n,k]^2,{k,0,n}],{n,0,20}] (* Harvey P. Dale, May 31 2015 *)

A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(xG(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n.

Original entry on oeis.org

1, 8, 24, 0, -168, 0, 2112, 0, -32040, 0, 536256, 0, -9542976, 0, 177126912, 0, -3390361128, 0, 66436117440, 0, -1326185205696, 0, 26872637815296, 0, -551301904867392, 0, 11428295231789568, 0, -239010764560888320, 0

Offset: 0

Paul D. Hanna, Feb 04 2010

sign

			G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 + ...
A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16*x)^(1/2)) is the power series:
G(x) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 + ... + C(2*n,n)^2*x^n + ...
The square root of g.f. A(x) begins:
A(x)^(1/2) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 + ... + A158101(n)*x^n + ...

Cf. A036917, A002894, A158101, A158100.

{a(n)=local(G=sum(m=0,n,binomial(2*m,m)^2*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}

{a(n)=if(n==1,8,polcoeff(agm(1,sqrt(1-16*x +x^2*O(x^n)))^(2*n-2),n)/(1-n))} \\ Paul D. Hanna, Mar 20 2010

G.f.: A(x) = x/Series_Reversion(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)^2*x^n = 1/agm(1, (1-16*x)^(1/2)) = g.f. of A002894 and G(x)^2 is the g.f. of A036917.
Self-convolution of A158101, which is a bisection of A158100; A158100 has g.f. F(x) that satisfies: F(x) = 1/AGM(1, 1 - 8*x/F(x) ).
a(n) = [x^n] AGM(1,(1-16*x)^(1/2))^(2*n-2)/(1-n) for n>1 where AGM is the arithmetic-geometric mean of Gauss. - Paul D. Hanna, Mar 20 2010

A300116 a(n) = Sum_{k=0..n} binomial(2k,k)^3 * binomial(2n-2k,n-k) * 2^(4*(n-k)).

Original entry on oeis.org

1, 40, 2008, 109120, 6173656, 357903040, 21090174400, 1257411781120, 75630327895000, 4580277582101440, 278915640538355008, 17061127317021130240, 1047543937631077672384, 64523332938885758410240, 3985152917145136901283328, 246717298245058901071237120

Offset: 0

Seiichi Manyama, Feb 25 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..554
Wadim Zudilin, Ramanujan-type formulae for 1/π: A second wind?, arXiv:0712.1332v2 [math.NT], 2008.

Cf. A036917, A380975.

{a(n) = sum(k=0, n, binomial(2*k, k)^3*binomial(2*n-2*k, n-k)*2^(4*(n-k)))}

n^3 * a(n) = 8 * (2*n-1) * (8*n^2-8*n+5) * a(n-1) - 4096 * (n-1)^3 * a(n-2) for n > 1.

a(n) ~ Gamma(1/4)^4 * 2^(6*n - 2) / (Pi^(7/2) * sqrt(n)). - Vaclav Kotesovec, Jul 10 2021

A089624 Expansion of sqrt(2/PiEllipticK(4sqrt(x))).

Original entry on oeis.org

1, 2, 16, 168, 1986, 25092, 330816, 4492560, 62352720, 879956000, 12583279360, 181872982400, 2652039363240, 38959845007440, 575974743052800, 8561706637619520, 127874111328349890, 1917875205285147780

Offset: 0

D. G. Rogers and Vladeta Jovovic, Dec 31 2003

nonn

When convolved with itself gives A002894.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A036917.

nmax = 20; CoefficientList[Series[Sqrt[Sum[Binomial[2*k, k]^2*x^k, {k, 0, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)
nmax = 20; CoefficientList[Series[Sqrt[2*EllipticK[16*x]/Pi], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)

{a(n) = if(n<0, 0, polcoeff( sqrt( sum(k=0, n, binomial(2*k, k)^2 * x^k, x*O(x^n)) ), n))} /* Michael Somos, Aug 17 2007 */

{a(n) = local(A); if(n<0, 0, A = x*O(x^n); polcoeff( subst( sum(k = 1, sqrtint(n), 2*x^k^2, 1+A), x, serreverse(x * (eta(x+A) * eta(x^4+A)^2 / eta(x^2+A)^3)^8 )), n))} /* Michael Somos, Aug 17 2007 */

Expansion of theta_3(q) in powers of (m/16) where q = exp(-Pi K'/K) and m = k^2 is the elliptic modulus. - Michael Somos, Aug 17 2007

a(n) ~ 2^(4*n-1) / (n*sqrt(Pi*log(n))) * (1 - (gamma/2 + 2*log(2)) / log(n) + (3*gamma^2/8 + 3*log(2)*gamma + 6*log(2)^2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

cp's OEIS Frontend

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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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).

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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.

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A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

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A036916 a(n) = Sum_{k=0..n} binomial(2*n-2*k,n-k)^2 * binomial(n,k)^2.

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A036915 a(n) = Sum_{k=0..n} C(2*n-2*k,n-k)^2 * C(2*n,k)^2.

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A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2*n,n)^2*x^n.

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).

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

A036916 a(n) = Sum_{k=0..n} binomial(2n-2k,n-k)^2 * binomial(n,k)^2.

A036915 a(n) = Sum_{k=0..n} C(2n-2k,n-k)^2 * C(2*n,k)^2.

A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(xG(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n.

A089624 Expansion of sqrt(2/PiEllipticK(4sqrt(x))).