A351288 -id:A351288 - cp's OEIS frontend

A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653

Offset: 0

Jeffrey Shallit

nonn

Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - R. H. Hardin, May 30 2008

Also, number of (n+1) X (n+1) binary symmetric matrices with zero main diagonal and rows, considered as binary numbers, in nondecreasing order. - Max Alekseyev, Feb 06 2022

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A008934, A060690, A089006, A351157, A351158, A351287, A351288.

Row sums of triangle A097712.

T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
a[n_] := Sum[T[n, k], {k, 0, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 02 2019 *)

@CachedFunction
def T(n, k): # T = A097712
    if k<0 or k>n: return 0
    elif k==0 or k==n: return 1
    else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A016121(n): return sum(T(n,k) for k in range(n+1))
[A016121(n) for n in range(31)] # G. C. Greubel, Feb 21 2024

a(n) = Sum_{k=0..n} A097712(n, k). - Paul D. Hanna, Aug 24 2004

Equals the binomial transform of A008934 (number of tournament sequences): a(n) = Sum_{k=0..n} C(n, k)*A008934(k). - Paul D. Hanna, Sep 18 2005

A351287 Number of symmetric 0-1 matrices with zero main diagonal and nondecreasing number of ones in the rows.

Original entry on oeis.org

1, 2, 4, 16, 84, 936, 16758, 602544, 37693734, 4588585904, 1016082688298, 436137488655846, 348748058993750616, 538461898813943437676

Offset: 1

Max Alekseyev, Feb 06 2022

Also, number of graphs with vertices labeled 1, 2, ..., n such that their degrees are nondecreasing.

Cf. A016121, A351157, A351288.

\\ See link in A295193 for GraphsByDegreeSeq.
a(n)={my(M=GraphsByDegreeSeq(n,n,(p,r)->1)); sum(i=1, matsize(M)[1], my(u=Vec(M[i,1])); prod(j=1, #u, u[j]!)*M[i,2]/n!)} \\ Andrew Howroyd, Feb 06 2022

def a351287(n): return sum(prod(factorial(e) for e in Partition((d+1 for d in G.degree_sequence())).to_exp()) // G.automorphism_group(return_group=False, order=True) for G in graphs(n))

a(11)-a(14) from Andrew Howroyd, Feb 06 2022

A351158 Number of n X n symmetric 0-1 matrices where rows are sorted lexicographically and row sums are nondecreasing.

Original entry on oeis.org

2, 5, 15, 56, 275, 1897, 18948

Offset: 1

Max Alekseyev, Feb 02 2022

Cf. A016121, A351157, A351287, A351288.

cp's OEIS Frontend

A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

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A351287 Number of symmetric 0-1 matrices with zero main diagonal and nondecreasing number of ones in the rows.

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A351158 Number of n X n symmetric 0-1 matrices where rows are sorted lexicographically and row sums are nondecreasing.

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