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 A331571 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order. 12
 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..209 FORMULA A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!. A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j). A(n, k) = 0 for k > 0, n > 2^(k-1). A331653(n) = Sum_{d|n} A(n/d, d). EXAMPLE Array begins: =============================================================== n\k | 0 1 2   3         4               5                 6 ----+----------------------------------------------------------   0 | 1 1 1   1         1               1                 1 ...   1 | 1 1 2   4         8              16                32 ...   2 | 1 0 3  23       290            4298             79143 ...   3 | 1 0 0 184     17488         2780752         689187720 ...   4 | 1 0 0 840    771305      1496866413     5261551562405 ...   5 | 1 0 0   0  21770070    585897733896 30607728081550686 ...   6 | 1 0 0   0 328149360 161088785679360 ...   ... The A(2,2) = 3 matrices are:    [1 1]  [1 0]  [1 0]    [1 0]  [1 1]  [0 1]    [0 1]  [0 1]  [1 1] PROG (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)} D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)} T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); } CROSSREFS Rows n=0..4 are A000012, A011782, A060090, A060491, A331652. Cf. A330942, A331567, A331569, A331570, A331572, A331653. Sequence in context: A102564 A215703 A292712 * A247504 A306800 A235955 Adjacent sequences:  A331568 A331569 A331570 * A331572 A331573 A331574 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Jan 20 2020 STATUS approved

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Last modified September 23 21:27 EDT 2020. Contains 337315 sequences. (Running on oeis4.)