A115257 -id:A115257 - cp's OEIS frontend

A079727 a(n) = 1 + C(2,1)^3 + C(4,2)^3 + ... + C(2n,n)^3.

1, 9, 225, 8225, 351225, 16354233, 805243257, 41229480825, 2172976383825, 117106008311825, 6423711336265041, 357470875526646609, 20131502573232075025, 1145190201805448075025, 65706503254247744075025

Offset: 0

Benoit Cloitre, Feb 17 2003

a(n) seems to have an interesting congruence property: For p prime, a(p) == 8 (mod p) if and only if p == 3, 5, 7, or 13 (mod 14); i.e., iff p = 7 or p is in A003625.
From Peter Bala, Jul 12 2024: (Start)
Zhi-Wei Sun (2010) conjectured that if p is an odd prime such that the Legendre symbol (p/7) = -1 (i.e., if p == 3, 5, 6 (mod 7)) then a(p-1) == 0 (mod p^2). Otherwise, if (p/7) = 1 then a(p-1) == 4*x^2 - 2*p (mod p^2) where p = x^2 + 7*y^2 with x, y in Z.
The author’s twin brother Zhi_Hong Sun confirmed the conjecture in the case (p/7) = -1.
Conjectures: if prime p is in A003625 then
1) a(p^2) == 8 + p^2 (mod p^3)
2) a(p*(p-1)) == p^2 (mod p^3)
3) a((p^2-1)/2) == p^2 (mod p^4) (all checked up to p = 101).
4) if n is a product of distinct primes from A003625 then a((n-1)/2) is divisible by n^2. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..556
Zhi-Hong Sun, Congruences concerning Legendre polynomials II", Theorem 3.2, arXiv:1012.3898v2 [math.NT], 2010-2012.
Zhi-Wei Sun, Open conjectures on congruences, Part A, conjecture A1, arXiv:0911.5665v59 [math.NT], 2009-2011.

Cf. A002476.

Cf. Sum_{k = 0..n} binomial(2*k, k)^m: A006134 (m=1), A115257 (m=2), this sequence (m=3).

[&+[Binomial(2*k, k)^3: k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Nov 16 2016

Table[Sum[Binomial[2 k, k]^3, {k, 0, n}], {n, 0, 14}] (* Michael De Vlieger, Nov 15 2016 *)

makelist(sum(binomial(2*k,k)^3,k,0,n),n,0,12); /* Emanuele Munarini, Nov 15 2016 */

PARI
```
a(n)=sum(k=0,n,binomial(2*k,k)^3)
```

a(n) = Sum_{k=0..n} binomial(2*k,k)^3.
G.f.: hypergeom([1/2, 1/2, 1/2], [1, 1], 64*x)/(1-x). - Vladeta Jovovic, Feb 18 2003
G.f.: hypergeom([1/4,1/4],[1],64*x)^2/(1-x). - Mark van Hoeij, Nov 17 2011
Recurrence: (n+2)^3*a(n+2)-(5*n+8)*(13*n^2+38*n+28)*a(n+1)+8*(2n+3)^3*a(n)=0. - Emanuele Munarini, Nov 15 2016
a(n) ~ 2^(6*n+6) / (63*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 16 2016

A115255 "Correlation triangle" of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 2, 6, 5, 6, 20, 14, 14, 20, 70, 46, 41, 46, 70, 252, 160, 134, 134, 160, 252, 924, 574, 466, 441, 466, 574, 924, 3432, 2100, 1672, 1534, 1534, 1672, 2100, 3432, 12870, 7788, 6118, 5506, 5341, 5506, 6118, 7788, 12870, 48620, 29172, 22692, 20152, 19174

Offset: 0

Paul Barry, Jan 18 2006

Row sums are A033114. Diagonal sums are A115256. T(2n,n) is A115257. Corresponds to the triangle of antidiagonals of the correlation matrix of the sequence array for C(2n,n).

Let s=(1,2,6,20,...), (central binomial coefficients), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A115255 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203005 for characteristic polynomials of principal submatrices of M, with interlacing zeros. - Clark Kimberling, Dec 27 2011

			Triangle begins:
  1;
  2, 2;
  6, 5, 6;
  20, 14, 14, 20;
  70, 46, 41, 46, 70;
  252, 160, 134, 134, 160, 252;
Northwest corner (square format):
  1    2    6    20    70
  2    5    14   46    160
  6    14   41   134   466
  20   46   134  441   1534

Cf. A203004, A203001, A202453.

s[k_] := Binomial[2 k - 2, k - 1];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]]; (* A115255 in square format *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A006134 *)
Table[m[1, j], {j, 1, 12}]     (* A000984 *)
Table[m[j, j], {j, 1, 12}]     (* A115257 *)
Table[m[j, j + 1], {j, 1, 12}] (* 2*A082578 *)
(* Clark Kimberling, Dec 27 2011 *)

G.f.: 1/(sqrt(1-4*x)*sqrt(1-4*x*y)*(1-x^2*y)) (format due to Christian G. Bower).

T(n, k) = Sum_{j=0..n} [j<=k]*C(2*k-2*j, k-j)*[j<=n-k]*C(2*n-2*k-2*j, n-k-j).

A228002 Alternate partial sums of binomial(2n,n)^2.

Original entry on oeis.org

1, 3, 33, 367, 4533, 58971, 794805, 10983819, 154653081, 2209251319, 31925528217, 465708778407, 6846750893929, 101325729466071, 1508015866093929, 22553429144856471, 338744206097695629, 5106973783924992771, 77251106929381097229, 1172036566162209342771

Offset: 0

Vaclav Kotesovec, Aug 07 2013

Seiichi Manyama, Table of n, a(n) for n = 0..833

Cf. A115257, A188679, A188680.

series(2*EllipticK(4*x^(1/2))/(Pi*(1+x)),x=0,20)

Table[Sum[(-1)^(n-k)*Binomial[2*k,k]^2, {k, 0, n}], {n, 0, 20}]

Recurrence: n^2*a(n) = (3*n-2)*(5*n-2)*a(n-1) + 4*(2*n-1)^2*a(n-2).

a(n) ~ 16^(n+1)/(17*Pi*n).

cp's OEIS Frontend

A079727 a(n) = 1 + C(2,1)^3 + C(4,2)^3 + ... + C(2n,n)^3.

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A115255 "Correlation triangle" of central binomial coefficients A000984.

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A228002 Alternate partial sums of binomial(2n,n)^2.

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