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A049401 Number of Young tableaux of height <= 5.

%I A049401 #59 Mar 27 2025 06:12:55
%S A049401 1,1,2,4,10,26,75,225,715,2347,7990,27908,99991,365587,1362310,
%T A049401 5159208,19831101,77233517,304423574,1212962072,4881181036,
%U A049401 19821471956,81165639197,334925706659,1391935877463,5823186349671,24511802558326,103772782048252,441696903185704
%N A049401 Number of Young tableaux of height <= 5.
%C A049401 Also the number of n-length words w over alphabet {a,b,c,d,e} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c) >= #(z,d) >= #(z,e), where #(z,x) counts the letters x in word z.  The a(5) = 26 words are: aaaaa, aaaab, aaaba, aabaa, abaaa, aaabb, aabab, abaab, aabba, ababa, aaabc, aabac, abaac, aabca, abaca, abcaa, aabbc, ababc, aabcb, abacb, abcab, aabcd, abacd, abcad, abcda, abcde. - _Alois P. Heinz_, May 30 2012
%D A049401 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), y_5(n), p. 452.
%H A049401 Alois P. Heinz, <a href="/A049401/b049401.txt">Table of n, a(n) for n = 0..1000</a>
%H A049401 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.
%H A049401 F. Bergeron and F. Gascon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/CYT/cyt.html">Counting Young tableaux of bounded height</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
%H A049401 Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, <a href="https://arxiv.org/abs/1708.00513">From Dyck paths to standard Young tableaux</a>, arXiv:1708.00513 [math.CO], 2017.
%H A049401 Dominique Gouyou-Beauchamps, <a href="https://doi.org/10.1016/S0195-6698(89)80034-4">Standard Young tableaux of height 4 and 5</a>, European J. Combin., 10(1):69-82, 1989.
%H A049401 Alon Regev, Amitai Regev, Doron Zeilberger, <a href="http://arxiv.org/abs/1507.03499">Identities in character tables of S_n</a>, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
%H A049401 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A049401 E.g.f.: e^x*(BesselI(0, 2*x)^2 - BesselI(0, 2*x)*BesselI(2, 2*x) - BesselI(0, 2*x)*BesselI(4, 2*x) - BesselI(1, 2*x)^2 + 2*BesselI(1, 2*x)*BesselI(3, 2*x) + BesselI(2, 2*x)*BesselI(4, 2*x) - BesselI(3, 2*x)^2) (BesselI = modified Bessel function of first kind).
%F A049401 a(n) ~ 3*5^(n+5)/(8*Pi*n^5). - _Vaclav Kotesovec_, Aug 18 2013
%F A049401 D-finite with recurrence (n+6)*(n+4)*a(n) +(-3*n^2-17*n-15)*a(n-1) -(13*n+9)*(n-1)*a(n-2) +15*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Sep 23 2021
%F A049401 a(n) = 6 * n! * Sum_{k=0..floor(n/2)} (2*k+2)!/((n-2*k)!*k!*(k+1)!*(k+2)!*(k+3)!). - _Seiichi Manyama_, Mar 27 2025
%p A049401 a:= proc(n) option remember;
%p A049401       `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*a(n-1)
%p A049401        +(n-1)*(13*n+9)*a(n-2) -15*(n-1)*(n-2)*a(n-3)) /
%p A049401        ((n+4)*(n+6)))
%p A049401     end:
%p A049401 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 12 2012
%t A049401 a[n_] := a[n] = If[n<3, {1, 1, 2}[[n+1]], ((3*n^2+17*n+15)*a[n-1] + (n-1)*(13*n+9)*a[n-2] - 15*(n-1)*(n-2)*a[n-3]) / ((n+4)*(n+6))]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 10 2014, after _Alois P. Heinz_ *)
%o A049401 (PARI) a(n) = 6*n!*sum(k=0, n\2, (2*k+2)!/((n-2*k)!*k!*(k+1)!*(k+2)!*(k+3)!)); \\ _Seiichi Manyama_, Mar 27 2025
%Y A049401 Sum of first five diagonals of A047884. Cf. A007579.
%Y A049401 Column k=5 of A182172. - _Alois P. Heinz_, May 30 2012
%K A049401 nonn,easy
%O A049401 0,3
%A A049401 _Alford Arnold_, _N. J. A. Sloane_
%E A049401 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001