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.

%I A016121 #29 Feb 22 2024 02:04:34
%S A016121 1,2,5,17,86,698,9551,226592,9471845,705154187,94285792211,
%T A016121 22807963405043,10047909839840456,8110620438438750647,
%U A016121 12062839548612627177590,33226539134943667506533207,170288915434579567358828997806,1630770670148598007261992936663653
%N 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.
%C A016121 Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - _R. H. Hardin_, May 30 2008
%C A016121 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
%H A016121 Alois P. Heinz, <a href="/A016121/b016121.txt">Table of n, a(n) for n = 0..50</a>
%F A016121 a(n) = Sum_{k=0..n} A097712(n, k). - _Paul D. Hanna_, Aug 24 2004
%F A016121 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
%t A016121 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}]]]];
%t A016121 a[n_] := Sum[T[n, k], {k, 0, n}];
%t A016121 a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 02 2019 *)
%o A016121 (SageMath)
%o A016121 @CachedFunction
%o A016121 def T(n, k): # T = A097712
%o A016121     if k<0 or k>n: return 0
%o A016121     elif k==0 or k==n: return 1
%o A016121     else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
%o A016121 def A016121(n): return sum(T(n,k) for k in range(n+1))
%o A016121 [A016121(n) for n in range(31)] # _G. C. Greubel_, Feb 21 2024
%Y A016121 Cf. A008934, A060690, A089006, A351157, A351158, A351287, A351288.
%Y A016121 Row sums of triangle A097712.
%K A016121 nonn
%O A016121 0,2
%A A016121 _Jeffrey Shallit_