A229068 - OEIS

%I #18 Feb 28 2022 01:52:40

%S 1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568503,2390466,

%T 10349340,46204720,211779200,997134592,4808141824,23745792032,

%U 119848119307,618058083314,3251373425356,17442275104496,95297400355320,530067682582320,2998503402985440

%N Number of standard Young tableaux of n cells and height <= 12.

%C Conjecture: generally (for tableaux with height <= k), a(n) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * Product_{j=1..k} Gamma(j/2); set k=12 for this sequence.

%H Vaclav Kotesovec and Alois P. Heinz, <a href="/A229068/b229068.txt">Table of n, a(n) for n = 0..400</a>

%F Recurrence: (n+11)*(n+20)*(n+27)*(n+32)*(n+35)*(n+36)*a(n) = 8*(14*n^5 + 1533*n^4 + 63328*n^3 + 1206473*n^2 + 10162354*n + 27335490)*a(n-1) + 16*(n-1)*(14*n^5 + 1372*n^4 + 49754*n^3 + 804616*n^2 + 5347213*n + 8718630)*a(n-2) - 128*(n-2)*(n-1)*(98*n^3 + 5391*n^2 + 90321*n + 438597)*a(n-3) - 256*(n-3)*(n-2)*(n-1)*(49*n^3 + 2343*n^2 + 32786*n + 121272)*a(n-4) + 110592*(n-4)*(n-3)*(n-2)*(n-1)*(2*n + 29)*a(n-5) + 147456*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n+12)*a(n-6).

%F a(n) ~ 602791875/128 * 12^(n+33)/(Pi^3*n^33).

%t RecurrenceTable[{-147456 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (12+n) a[-6+n]-110592 (-4+n) (-3+n) (-2+n) (-1+n) (29+2 n) a[-5+n]+256 (-3+n) (-2+n) (-1+n) (121272+32786 n+2343 n^2+49 n^3) a[-4+n]+128 (-2+n) (-1+n) (438597+90321 n+5391 n^2+98 n^3) a[-3+n]-16 (-1+n) (8718630+5347213 n+804616 n^2+49754 n^3+1372 n^4+14 n^5) a[-2+n]-8 (27335490+10162354 n+1206473 n^2+63328 n^3+1533 n^4+14 n^5) a[-1+n]+(11+n) (20+n) (27+n) (32+n) (35+n) (36+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10, a[5]==26, a[6]==76}, a, {n, 20}]

%Y Cf. A182172, A001405 (k=2), A001006 (k=3), A005817 (k=4), A049401 (k=5), A007579 (k=6), A007578 (k=7), A007580 (k=8), A212915 (k=9), A212916 (k=10), A229053 (k=11).

%Y Column k=12 of A182172.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Sep 12 2013