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

%I A129123 #33 Mar 25 2025 12:44:09
%S A129123 1,1,2,17,98,882,7812,78129,815474,8955650,101869508,1194964498,
%T A129123 14374530436,176681194276,2212121332488,28145258688369,
%U A129123 363177582488274,4745064935840178,62687665026816228,836447728509168930,11261240896657686660,152847558411986548260
%N A129123 Number of 4-tuples of standard tableau with height less than or equal to 2.
%C A129123 Number of pairs of Dyck paths of semilength n with equal midpoint. - _Alois P. Heinz_, Oct 07 2022
%H A129123 Alois P. Heinz, <a href="/A129123/b129123.txt">Table of n, a(n) for n = 0..839</a>
%H A129123 F. Bergeron, L. Favreau and D. Krob, <a href="http://dx.doi.org/10.1016/0012-365X(94)00148-C">Conjectures on the enumeration of tableaux of bounded height</a>, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
%F A129123 a(n) = Sum_{k=0..n} A120730(n,k)^4. - _Philippe Deléham_, Oct 18 2008
%F A129123 From _Vaclav Kotesovec_, Dec 16 2017: (Start)
%F A129123 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).
%F A129123 a(n) ~ 3* 2^(4*n - 1/2) / (Pi^(3/2) * n^(7/2)). (End)
%F A129123 a(n) = A357652(n) - A355481(n). - _Alois P. Heinz_, Oct 13 2022
%F A129123 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
%F A129123 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^3. - _Seiichi Manyama_, Mar 25 2025
%p A129123 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A129123      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A129123     end:
%p A129123 a:= n-> add(b(n, n-2*j)^4, j=0..n/2):
%p A129123 seq(a(n), n=0..21);  # _Alois P. Heinz_, Mar 25 2025
%t A129123 Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^4, {k,0, Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Dec 16 2017 *)
%o A129123 (PARI) a(n) = sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^4);
%o A129123 (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
%o A129123 (SageMath)
%o A129123 def A129123(n): return sum(((n-2*j+1)/(n-j+1))^4*binomial(n,j)^4 for j in range((n//2)+1))
%o A129123 [A129123(n) for n in range(31)] # _G. C. Greubel_, Nov 08 2022
%o A129123 (Python)
%o A129123 from math import comb
%o A129123 def A129123(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1)) # _Chai Wah Wu_, Mar 25 2025
%Y A129123 Cf. A000108, A001405, A003161, A120730, A355481, A357652.
%Y A129123 Column k=4 of A357824.
%K A129123 nonn
%O A129123 0,3
%A A129123 _Mike Zabrocki_, Mar 29 2007