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 A333901 Array read by antidiagonals: T(n,k) is the number of n X k nonnegative integer matrices with all column sums n and row sums k. 11
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 55, 19, 1, 1, 1, 1, 51, 415, 415, 51, 1, 1, 1, 1, 141, 3391, 10147, 3391, 141, 1, 1, 1, 1, 393, 28681, 261331, 261331, 28681, 393, 1, 1, 1, 1, 1107, 248137, 7100821, 22069251, 7100821, 248137, 1107, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS T(n,k) is the number of ordered factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity). LINKS Andrew Howroyd, Table of n, a(n) for n = 0..405 (antidiagonals n=0..27) FORMULA T(n,k) = T(k,n). EXAMPLE Array begins: ======================================================= n\k | 0 1 2 3 4 5 6 ----+-------------------------------------------------- 0 | 1 1 1 1 1 1 1 ... 1 | 1 1 1 1 1 1 1 ... 2 | 1 1 3 7 19 51 141 ... 3 | 1 1 7 55 415 3391 28681 ... 4 | 1 1 19 415 10147 261331 7100821 ... 5 | 1 1 51 3391 261331 22069251 1985311701 ... 6 | 1 1 141 28681 7100821 1985311701 602351808741 ... ... The T(3,2) = 7 matrices are: [1 1] [1 1] [1 1] [2 0] [2 0] [0 2] [0 2] [1 1] [2 0] [0 2] [1 1] [0 2] [1 1] [2 0] [1 1] [0 2] [2 0] [0 2] [1 1] [2 0] [1 1] PROG (PARI) T(n, k)={ local(M=Map(Mat([k, 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, 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], 0, src[i, 2], k))); vecsum(Mat(M)[, 2]) } for(n=0, 7, for(k=0, 7, print1(T(n, k), ", ")); print) CROSSREFS Columns k=0..9 are A000012, A000012, A002426, A172743, A172816, A172868, A172904, A172931, A172947, A172961. Main diagonal is A110058. Cf. A257462, A257493. Sequence in context: A147989 A119329 A334549 * A054724 A360440 A349350 Adjacent sequences: A333898 A333899 A333900 * A333902 A333903 A333904 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Apr 18 2020 STATUS approved

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Last modified September 16 23:59 EDT 2024. Contains 375984 sequences. (Running on oeis4.)