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A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A214775 #40 Mar 30 2023 11:03:08
%S A214775 1,1,1,2,6,2,5,25,25,5,14,98,174,98,14,42,378,962,962,378,42,132,1452,
%T A214775 4804,7020,4804,1452,132,429,5577,22689,43573,43573,22689,5577,429,
%U A214775 1430,21450,103510,245962,325590,245962,103510,21450,1430
%N A214775 Number T(n,k) of solid standard Young tableaux of shape [[n,k],[n-k]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A214775 T(n,k) is odd if and only if n = 2^i-1 for i in {0, 1, 2, ... } = A001477.
%H A214775 Alois P. Heinz, <a href="/A214775/b214775.txt">Rows n = 0..140, flattened</a>
%H A214775 S. B. Ekhad, D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229v1 [math.CO], 2012
%H A214775 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A214775 Triangle T(n,k) begins:
%e A214775     1;
%e A214775     1,    1;
%e A214775     2,    6,    2;
%e A214775     5,   25,   25,    5;
%e A214775    14,   98,  174,   98,   14;
%e A214775    42,  378,  962,  962,  378,   42;
%e A214775   132, 1452, 4804, 7020, 4804, 1452, 132;
%e A214775   ...
%p A214775 b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y),
%p A214775       `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
%p A214775       `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0)))
%p A214775     end:
%p A214775 T:= (n, k)-> b(n, k, n-k):
%p A214775 seq(seq(T(n, k), k=0..n), n=0..10);
%t A214775 b[x_, y_, z_] := b[x, y, z] = If[z>y, b[x, z, y], If[Union[{x, y, z}] == {0}, 1, If[x>y && x>z, b[x-1, y, z], 0] + If[y>0, b[x, y-1, z], 0] + If[z>0, b[x, y, z-1], 0]]]; T[n_, k_] := b[n, k, n-k]; Table[T[n, k] , {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 15 2014, translated from Maple *)
%o A214775 (Sage)
%o A214775 @CachedFunction
%o A214775 def B(x, y, z) :
%o A214775     if z > y : return B(x, z, y)
%o A214775     if x==y and y==z and z==0 : return 1
%o A214775     a = B(x-1, y, z) if x>y and x>z else 0
%o A214775     b = B(x, y-1, z) if y>0 else 0
%o A214775     c = B(x, y, z-1) if z>0 else 0
%o A214775     return a + b + c
%o A214775 T = lambda n, k: B(n, k, n-k)
%o A214775 [[T(n, k) for k in (0..n)] for n in (0..10)]
%o A214775 # After Maple code of Alois P. Heinz. _Peter Luschny_, Jul 30 2012
%Y A214775 Columns 0-5 give: A000108, A214955, A215298, A215299, A215300, A215301.
%Y A214775 Row sums give: A215002.
%Y A214775 Central row elements give: A214801.
%K A214775 nonn,tabl
%O A214775 0,4
%A A214775 _Alois P. Heinz_, Jul 28 2012