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 A333157 Triangle read by rows: T(n,k) is the number of n X n symmetric binary matrices with k ones in every row and column. 15
 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 18, 10, 1, 1, 26, 112, 112, 26, 1, 1, 76, 820, 1760, 820, 76, 1, 1, 232, 6912, 35150, 35150, 6912, 232, 1, 1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1, 1, 2620, 708256, 24243520, 133948836, 133948836, 24243520, 708256, 2620, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is the number of k-regular symmetric relations on n labeled nodes. T(n,k) is the number of k-regular graphs with half-edges on n labeled vertices. Terms may be computed without generating all graphs by enumerating the number of graphs by degree sequence. A PARI program showing this technique is given below. Burnside's lemma as applied in A122082 and A000666 can be used to extend this method to the case of unlabeled vertices A333159 and A333161 respectively. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..230 FORMULA T(n,k) = T(n,n-k). EXAMPLE Triangle begins:   1,   1,   1;   1,   2,     1;   1,   4,     4,      1;   1,  10,    18,     10,       1;   1,  26,   112,    112,      26,      1;   1,  76,   820,   1760,     820,     76,     1;   1, 232,  6912,  35150,   35150,   6912,   232,   1;   1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1;   ... PROG (PARI) \\ See script in A295193 for comments. GraphsByDegreeSeq(n, limit, ok)={   local(M=Map(Mat([x^0, 1])));   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));   my(recurse(r, p, i, q, v, e) = if(e<=limit && poldegree(q)<=limit, if(i<0, if(ok(x^e+q, r), acc(x^e+q, v)), my(t=polcoeff(p, i)); for(k=0, t, self()(r, p, i-1, (t-k+x*k)*x^i+q, binomial(t, k)*v, e+k)))));   for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0))); Mat(M); } Row(n)={my(M=GraphsByDegreeSeq(n, n\2, (p, r)->poldegree(p)-valuation(p, x) <= r + 1), v=vector(n+1)); for(i=1, matsize(M)[1], my(p=M[i, 1], d=poldegree(p)); v[1+d]+=M[i, 2]; if(pollead(p)==n, v[2+d]+=M[i, 2])); for(i=1, #v\2, v[#v+1-i]=v[i]); v} for(n=0, 8, print(Row(n))) \\ Andrew Howroyd, Mar 14 2020 CROSSREFS Columns k=0..8 are A000012, A000085, A000986, A110040, A139670, A139671, A139673, A139674, A139675. Row sums are A322698. Central coefficients are A333164. Cf. A188448 (transposed as array). Cf. A059441, A295193, A333158, A333159, A333161. Sequence in context: A326326 A307139 A078121 * A119732 A260625 A306614 Adjacent sequences:  A333154 A333155 A333156 * A333158 A333159 A333160 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Mar 09 2020 STATUS approved

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)