A129123 Number of 4-tuples of standard tableau with height less than or equal to 2.

1, 1, 2, 17, 98, 882, 7812, 78129, 815474, 8955650, 101869508, 1194964498, 14374530436, 176681194276, 2212121332488, 28145258688369, 363177582488274, 4745064935840178, 62687665026816228, 836447728509168930, 11261240896657686660, 152847558411986548260

Offset

0,3

Comments

Number of pairs of Dyck paths of semilength n with equal midpoint. - Alois P. Heinz, Oct 07 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 0..839

F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

Formula

a(n) = Sum_{k=0..n} A120730(n,k)^4. - Philippe Deléham, Oct 18 2008

From Vaclav Kotesovec, Dec 16 2017: (Start)

Recurrence: n*(n+1)^3*(15*n^2 - 34*n + 7)*a(n) = 2*n*(90*n^5 - 309*n^4 + 147*n^3 + 124*n^2 - 135*n + 35)*a(n-1) + 4*(n-1)^2*(4*n - 5)*(4*n - 3)*(15*n^2 - 4*n - 12)*a(n-2).

a(n) ~ 3* 2^(4*n - 1/2) / (Pi^(3/2) * n^(7/2)). (End)

a(n) = A357652(n) - A355481(n). - Alois P. Heinz, Oct 13 2022

a(n) = Sum_{j=0..floor(n/2)} ((n-2*j+1)/(n-j+1))^4 * binomial(n,j)^4. - G. C. Greubel, Nov 08 2022

Mathematica

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^4, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2017 *)

Prog

(PARI) a(n)=sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^4)
(Magma) [(&+[((n-2*j+1)/(n-j+1))^4*Binomial(n, j)^4: j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Nov 08 2022
(SageMath)
def A129123(n): return sum(((n-2*j+1)/(n-j+1))^4*binomial(n, j)^4 for j in range((n//2)+1))
[A129123(n) for n in range(31)] # G. C. Greubel, Nov 08 2022

Crossrefs

Cf. A000108, A001405, A003161, A120730, A355481, A357652.

Column k=4 of A357824.

Sequence in context: A002645 A100268 A163790 * A109724 A127533 A023260

Adjacent sequences: A129120 A129121 A129122 * A129124 A129125 A129126

Keyword

nonn

Author

Mike Zabrocki, Mar 29 2007

Status

approved