A338403
Regular triangle read by rows: T(n,k) is the number of (n,k)-Duck words, for n>=1 and 0<=k<=n-1.
Original entry on oeis.org
1, 2, 3, 5, 23, 14, 14, 131, 233, 84, 42, 664, 2339, 2367, 594, 132, 3166, 18520, 36265, 24714, 4719, 429, 14545, 127511, 408311, 527757, 266219, 40898
Offset: 1
Triangle begins:
1;
2, 3;
5, 23, 14;
14, 131, 233, 84;
42, 664, 2339, 2367, 594;
...
A353173
Dimension of space of invariants of n-th tensor power of the 26-dimensional fundamental (or "standard") irreducible representation of F_4.
Original entry on oeis.org
1, 0, 1, 1, 5, 15, 70, 330, 1820, 10858, 70875, 497135, 3727955, 29658410, 248989676, 2194891440, 20231692430, 194286848280, 1937546532820, 20008993160460, 213436182918652, 2346406693816315, 26531060178217182, 307987244037724262, 3664579007885995952
Offset: 0
a(1)=0 because there is no F_4-invariant linear form on the 26-dimensional representation; a(2)=1 because there is, up to scalars, precisely one invariant quadratic form.
The analogous sequence for the (52-dimensional) adjoint representation of F_4 is:
A179685.
A similar sequence for G_2 (for its 7-dimensional fundamental irreducible representation) is:
A059710.
A similar sequence for B_2 (for its standard 5-dimensional irreducible representation) is:
A095922.
For A_n the similar sequence (omitting some 0's) is given by the (n+1)-dimensional Catalan numbers, e.g.,
A005789 for A_2.
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p_tensor(n,[0,0,0,1],F4)|[0,0,0,0] # Returns the value of a(n).
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