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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.
17
1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653
OFFSET
0,2
COMMENTS
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
LINKS
FORMULA
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
MATHEMATICA
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 *)
PROG
(SageMath)
@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
CROSSREFS
Row sums of triangle A097712.
Sequence in context: A162041 A162042 A162043 * A216519 A245108 A195137
KEYWORD
nonn
STATUS
approved