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A120933 Triangle read by rows: T(n,k) is the number of binary words of length n for which the length of the maximal leading nondecreasing subword is k (1<=k<=n).

%I A120933 #7 Sep 03 2020 12:19:16
%S A120933 2,1,3,2,2,4,4,4,3,5,8,8,6,4,6,16,16,12,8,5,7,32,32,24,16,10,6,8,64,
%T A120933 64,48,32,20,12,7,9,128,128,96,64,40,24,14,8,10,256,256,192,128,80,48,
%U A120933 28,16,9,11,512,512,384,256,160,96,56,32,18,10,12,1024,1024,768,512,320
%N A120933 Triangle read by rows: T(n,k) is the number of binary words of length n for which the length of the maximal leading nondecreasing subword is k (1<=k<=n).
%H A120933 Thomas Grubb and Frederick Rajasekaran, <a href="https://arxiv.org/abs/2009.00650">Set Partition Patterns and the Dimension Index</a>, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.
%F A120933 T(n,k) = k*2^(n-k-1) if k<n; T(n,n) = n+1.
%F A120933 G.f.: G(t,z) = (1-2z+tz^2)/[(1-2z)(1-tz)^2] - 1.
%F A120933 Row sums are the powers of 2 (A000079).
%F A120933 Sum_{k=1..n} k*T(n,k) = 3*2^n-n-3 = A095151(n).
%e A120933 T(4,2)=4 because we have 0100,0101,1100 and 1101.
%e A120933 Triangle starts:
%e A120933 2;
%e A120933 1,3;
%e A120933 2,2,4;
%e A120933 4,4,3,5;
%e A120933 8,8,6,4,6;
%p A120933 T:=proc(n,k) if k<n then k*2^(n-k-1) elif k=n then n+1 else 0 fi end: for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form;
%Y A120933 Cf. A000079, A095151.
%K A120933 nonn,tabl
%O A120933 1,1
%A A120933 _Emeric Deutsch_, Jul 16 2006