A195644 T(n,k) is the number of lower triangles of an n X n 0..k array with all row sums equal to the length of the row and all column sums equal to the length of the column.
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 15, 1, 1, 1, 3, 19, 199, 1, 1, 1, 3, 19, 379, 6247, 1, 1, 1, 3, 19, 391, 22506, 505623, 1, 1, 1, 3, 19, 391, 25428, 4063437, 105997283, 1, 1, 1, 3, 19, 391, 25532, 5422820, 2303397986, 58923059879, 1, 1, 1, 3, 19, 391, 25532, 5536654
Offset: 1
Examples
Table starts: 1 1 1 1 1 1 ... 1 1 1 1 1 1 ... 1 3 3 3 3 3 ... 1 15 19 19 19 19 ... 1 199 379 391 391 391 ... 1 6247 22506 25428 25532 25532 ... 1 505623 4063437 5422820 5536654 5539434 ... 1 105997283 2303397986 3868544673 4102276124 4116036800 ... ... Some solutions for n=5, k=4: ..1..........1..........1..........1..........1..........1..........1 ..1.1........1.1........2.0........2.0........2.0........2.0........2.0 ..3.0.0......3.0.0......0.1.2......2.1.0......0.1.2......2.0.1......1.2.0 ..0.1.1.2....0.1.1.2....1.1.0.2....0.1.1.2....1.1.0.2....0.0.2.2....1.2.1.0 ..0.2.2.0.1..0.2.2.0.1..1.2.1.0.1..0.2.2.0.1..1.2.1.0.1..0.4.0.0.1..0.0.2.2.1
Links
- R. H. Hardin, Table of n, a(n) for n = 1..102
Crossrefs
Programs
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PARI
\\ adapted from program for A257493. T(n, k)={ local(M=Map(Mat([0, 1]))); my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v))); my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, min(k, h-i), for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m))))); for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1] + x^(r-1), 0, src[i, 2], r))); vecsum(Mat(M)[, 2]) } \\ Andrew Howroyd, May 16 2020
Formula
T(n,k) = T(n,k-1) for k >= n, n >= 2. - Andrew Howroyd, May 16 2020