A183204 -id:A183204 - cp's OEIS frontend

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)!.

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).

A245086 Central values of the n-th discrete Chebyshev polynomials of order 2n.

Original entry on oeis.org

1, 0, -6, 0, 90, 0, -1680, 0, 34650, 0, -756756, 0, 17153136, 0, -399072960, 0, 9465511770, 0, -227873431500, 0, 5550996791340, 0, -136526995463040, 0, 3384731762521200, 0, -84478098072866400, 0, 2120572665910728000, 0, -53494979785374631680, 0

Offset: 0

Nikita Gogin, Jul 11 2014

In the general case the n-th discrete Chebyshev polynomial of order N is D(N,n;x) = Sum_{i = 0..n} (-1)^i*C(n,i)*C(N-x,n-i)*C(x,i). For N = 2*n , x = n, one gets a(n) = D(2n,n;n) = Sum_{i = 0..n} (-1)^i*C(n,i)^3 that equals (due to Dixon's formula) 0 for odd n and (-1)^m*(3m)!/(m!)^3 for n = 2*m. (Riordan, 1968) So, a(2*m) = (-1)^m*A006480(m).

John Riordan, Combinatorial Identities, John Willey&Sons Inc., 1968.

Nikita Gogin and Mika Hirvensalo, On the Generating Function of Discrete Chebyshev Polynomials, TUCS Technical Reports 819, Turku Centre for Computer Science, 2007.
Nikita Gogin and Mika Hirvensalo, Recurrent Constructions of the MacWilliams and Chebyshev Matrices, Fundamenta Informaticae, vol.116, issue 1-4, January 2012, pages 93-110.
H. W. Gould, Tables of Combinatorial Identities, Edited by J. Quaintance.
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
Y. Sun and X. Wang, A new proof of a curious identity, Math Gazette, Vol. 91, No. 520 (Mar 2007) 105-107.
Wikipedia, Dixon's identity

Cf. A000172, A002894, A005260, A006480, A103882, A183204, A273630, A364303.

Table[Coefficient[Simplify[JacobiP[n,0,-(2*n+1),(1+t^2)/(1-t^2)]*(1-t^2)^n],t,n],{n,0,20}]

from math import factorial
def A245086(n): return 0 if n&1 else (-1 if (m:=n>>1)&1 else 1)*factorial(3*m)//factorial(m)**3 # Chai Wah Wu, Oct 04 2022

a(n) is a coefficient at t^n in (1-t^2)^n*P(0,-(2*n+1);n;(1+t^2)/(1-t^2)), where P(a,b;k;x) is the k-th Jacobi polynomial (Gogin and Hirvensalo, 2007).
G.f.: Hypergeometric2F1[1/3,2/3,1,-27*x^2].
a(2*m+1) = 0, a(2*m) = (-1)^m*A006480(m).
From Peter Bala, Aug 04 2016: (Start)
a(n) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*binomial(2*n - k,n)*binomial(n + k,n) (Sun and Wang).
a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k, n - k)*binomial(2*k, k)*binomial(2*n - k, n) (Gould, Vol.5, 9.23).
a(n) = -1/(n + 1)^3 * A273630(n+1). (End)
From Peter Bala, Mar 22 2022: (Start)
a(n) = - (3*(3*n-2)*(3*n-4)/n^2)*a(n-2).
a(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A002894(n) = binomial(2*n,n)^2 = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)). Cf. A103882. (End)
From Peter Bala, Jul 23 2023: (Start)
a(n) = [x^n] G(x)^(3*n), where the power series G(x) = 1 - x^2 + 2*x^4 - 14*x^6 + 127*x^8 - 1364*x^10 + ... appears to have integer coefficients.
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^3, where the power series F(x) = 1 - x^2  + 8*x^4 - 101*x^6 + 1569*x^8 - 27445*x^10 + ..., appears to have integer coefficients. See A229452.
Row 1 of A364303. (End)
a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(n+k, k)^2 * binomial(3*n+1, n-k). Cf. A183204.- Peter Bala, Sep 20 2024

A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3x^k](1-x)^(3n+1), read by rows of k=0..2n terms.

Original entry on oeis.org

1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 54, 405, 760, 405, 54, 1, 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1, 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1, 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1, 1, 490, 35623, 818300, 7917371, 37215794, 91789005, 123519792, 91789005, 37215794, 7917371, 818300, 35623, 490, 1, 1, 704, 73200, 2430400, 34657700, 246781248, 955910032, 2116980800, 2751843600, 2116980800, 955910032, 246781248, 34657700, 2430400, 73200, 704, 1

Offset: 0

Paul D. Hanna, Oct 30 2010

			Triangle begins:
 1;
 1, 4, 1;
 1, 20, 48, 20, 1;
 1, 54, 405, 760, 405, 54, 1;
 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1;
 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1;
 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1; ...
Row g.f.s begin:
 (1) = (1-x)*(1 + x + x^2 + x^3 + x^4 +...);
 (1 + 4*x + x^2) = (1-x)^4*(1 + 2^3*x + 3^3*x^2 + 4^3*x^3 +...);
 (1 + 20*x + 48*x^2 + 20*x^3 + x^4) = (1-x)^7*(1 + 3^3*x + 6^3*x^2 +...);
 (1 + 54*x + 405*x^2 + 760*x^3 + 405*x^4 + 54*x^5 + x^6) = (1-x)^10*(1 + 4^3*x + 10^3*x^2 + 20^3*x^3 + 35^3*x^4 +...); ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, as a flattened triangle of rows 0..32
Ilia Gaiur, Vladimir Rubtsov, and Duco van Straten, Product formulas for the Higher Bessel functions, arXiv:2405.03015 [math.AG], 2024. See p. 18.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.

Cf. A181543, A181545, A006480, A002897, A000172.

Cf. A183204 (central terms), A183205.

t[n_, k_] := SeriesCoefficient[Sum[Binomial[n+j, j]^3*x^j, {j, 0, n+k}]*(1-x)^(3*n+1), {x,0, k}]; Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Feb 04 2014, after PARI *)

{T(n,k)=polcoeff(sum(j=0,n+k,binomial(n+j,j)^3*x^j)*(1-x)^(3*n+1),k)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

Row sums equal A006480(n) = (3n)!/(n!)^3, which is de Bruijn's s(3,n).
From Yahia Kahloune, Jan 30 2014: (Start)
Using these coefficients we can obtain formulas for the sums
Sum_{i=1..n} C(e-1+i,e)^3. Let us define b(k,e,3) = sum_{i=0..k-e} (-1)^i*C(3*e+1,i)*C(k-i,e)^3, where k=e+i.
For example:
b(e,e,3) = 1;
b(e+1,e,3) = (e+1)^3-(3*e+1) = e^2*(e+3);
b(e+2,e,3) = C(e+2,2)^3 - (3*e+1)*(e+1)^3 + C(3*e+1,2);
b(e+3,e,3) = C(e+3,e)^3 - (3*e+1)*C(e+2,e)^3 + C(3*e+1,2)*C(e+1,e)^3 - C(3*e+1,3);
b(e+4,e,3) = C(e+4,e)^3 - (3*e+1)*C(e+3,e)^3  + C(3*e+1,2)*C(e+2,e) - C(3*e+1,3)*C(e+1,e)^3 + C(3*e+1,4).
Then we have the formula: Sum_{i=1..n} C(e-1+i,e)^3 = Sum_{i=0..2*e} b(e+i,e,3)*C(n+e+i,3*e+1).
Example: Sum_{i=1..7} C(2+i,3)^3 = C(10,10) + 54*C(11,10) + 405*C(12,10) + 760*C(13,10) + 405*C(14,10) + 54*C(15,10) + C(16,10) = 820260. (End)
Let E be the operator D*x*D*x*D, where D denotes the derivative operator d/dx. Then (1/(n)!^3)  * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(3*n+1) = Sum_{k >= 0} binomial(n+k, k)^3*x^k. For example, when n = 2 we have (1/2!)^3*E^3(1/(1 - x)) = (1 + 20 x + 48 x^2 + 20 x^3 + x^4)/(1 - x)^7. - Sergii Voloshyn, Dec 03 2024

A376458 a(n) = Sum_{k = 0..n} (-1)^(n+k)binomial(n, k)binomial(n+k, k)*A108625(n-1, n-k).

Original entry on oeis.org

1, 1, -7, 1, 569, -3749, -45151, 806737, 1052729, -130060889, 740060243, 16076432923, -238772815711, -1050791121197, 49401000432497, -171944622257999, -7658491447803847, 87632552103603679, 768037618172427023, -22023427875902878553, 19183786570616924819, 4030690809877385503081, -33792039667279104716677, -520860578851790657166869

Offset: 0

Peter Bala, Sep 23 2024

Compare with the following identity relating the sequence of Apéry numbers to the table of crystal ball sequences for the A_n lattices: A005259(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n, k), which can be verified by using the MulZeil procedure in Zeilberger's MultiZeilberger Maple package to find a recurrence for the double sum on the right-hand side of the identity.
The sequence of Apéry numbers A005259 satisfies the supercongruences A005259(n*p^r) == A005259(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r.
We conjecture that the present sequence satisfies same supercongruences and in addition satisfies the stronger congruences a(p) == a(1) (mod p^5) for all primes p >= 7 (checked up to p = 199) and a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) for all primes p >= 5 and integers r >= 2.

			Examples of supercongruences:
a(7) - a(1) = 806737 - 1 = (2^4)*3*(7^5) == 0 (mod 7^5).
a(11) - a(1) = 16076432923 - 1 =  2*3*(11^5)*127*131 == 0 (mod 11^5).
a(5^2) - a(5) = 22511570786292886382808751 - (-3749) = (2^2)*(3^2)*(5^9)*67*97*
7741*49223*129289 == 0 (mod 5^9).

Cf. A005259, A108625, A183204, A352653, A376459 - A376466.

A108625(n, k) := add(binomial(n, i)^2 * binomial(n+k-i, k-i), i = 0..k):
a(n) := add((-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n-1, n-k), k = 0..n):
seq(a(n), n = 0..25);

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

P-recursive: (2*n - 3)*n^3*(n - 1)^2*(473*n^5 - 4988*n^4 + 20888*n^3 - 43462*n^2 + 45019*n - 18634)*a(n) = - 2*(n - 1)^2*(3784*n^9 - 51256*n^8 + 303801*n^7 - 1037327*n^6 + 2252744*n^5 - 3220636*n^4 + 3006247*n^3 - 1739455*n^2 + 555714*n - 75024)*a(n-1) - 2*(n - 2)*(2*n - 1)*(52030*n^9 - 756800*n^8 + 4787337*n^7 - 17271387*n^6 + 39143817*n^5 - 57806236*n^4 + 55708921*n^3 - 33926177*n^2 + 11955879*n - 1890360)*a(n-2) - 2*(n - 2)*(n - 3)^3*(2*n - 1)*(2*n - 3)*(473*n^5 - 2623*n^4 + 5666*n^3 - 5996*n^2 + 3172*n - 704)*a(n-3) with a(0) = 1, a(1) = -1 and a(2) = 7.

A183205 a(n) = [x^n] (1-x)^(3n+1)/(n+1) * Sum_{k>=0} C(n+k-1,k)^3*x^k.

Original entry on oeis.org

1, 2, 16, 190, 2768, 45584, 814728, 15439974, 305760400, 6265985440, 131980086368, 2843029539376, 62400628835608, 1391503990134080, 31454839290752912, 719470742267557110, 16627360903974831120, 387786053931422003360

Offset: 0

Paul D. Hanna, Dec 30 2010

nonn

Cf. A181544, A183204.

{a(n)=polcoeff((1-x)^(3*n+1)/(n+1)*sum(j=0, 2*n, binomial(n+j, j)^3*x^j), n)}

a(n) = A183204(n)/(n+1), where A183204 equals the central terms of triangle A181544.

A279613 Expansion of the g.f. of A160534 in powers of A121593.

Original entry on oeis.org

1, -7, 42, -231, 1155, -4998, 15827, -791, -566244, 6506955, -53524611, 369879930, -2218053747, 11306008875, -43772711220, 55203364377, 1172838094533, -16542312772356, 150992704165079, -1130142960861845, 7290759457923816

Offset: 1

Lynette O'Brien, Dec 15 2016

sign

(eta(q))^7/eta(7*q) in powers of (eta(7*q)/eta(q))^4.

This sequence is u_n in Theorem 6.5 in O'Brien's thesis.

			G.f.: 1 - 7*x + 42*x^2 - 231*x^3 + 1155*x^4 - 4998*x^5 + ...

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

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

Cf. A183204, A229111, A279618, A279619.

(n+1)^4a_7(n+1)=-(26*n^4+52*n^3+58*n^2+32*n+7)a_7(n)-(267*n^4+268*n^2+18)a_7(n-1)-(1274*n^4-2548*n^3+2842*n^2-1568*n+343)a_7(n-2)-2401(n-1)^4a_7(n-3)
  with a_7(0)=1, a_7(-1)=a_7(-2)=a_7(-3)=0.
asymptotic conjecture: a(n) ~ C n^(-4/3) 7^n cos( n( arctan( (3*sqrt 3)/13) +Pi -1.083913253)), where C = 6.502807770...

cp's OEIS Frontend

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)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A245086 Central values of the n-th discrete Chebyshev polynomials of order 2n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3*x^k]*(1-x)^(3n+1), read by rows of k=0..2n terms.

Views

Author

Keywords

Examples

Links

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).

A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3x^k](1-x)^(3n+1), read by rows of k=0..2n terms.

A376458 a(n) = Sum_{k = 0..n} (-1)^(n+k)binomial(n, k)binomial(n+k, k)*A108625(n-1, n-k).