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A087322 Triangle T read by rows: T(n, 1) = 2*n + 1. For 1 < k <= n, T(n, k) = 2*T(n,k-1) + 1.

%I A087322 #29 Oct 17 2023 05:01:21
%S A087322 3,5,11,7,15,31,9,19,39,79,11,23,47,95,191,13,27,55,111,223,447,15,31,
%T A087322 63,127,255,511,1023,17,35,71,143,287,575,1151,2303,19,39,79,159,319,
%U A087322 639,1279,2559,5119,21,43,87,175,351,703,1407,2815,5631,11263,23,47,95
%N A087322 Triangle T read by rows: T(n, 1) = 2*n + 1. For 1 < k <= n, T(n, k) = 2*T(n,k-1) + 1.
%C A087322 With T(n,0) = n for n >= 0, this becomes _J. M. Bergot_'s triangular array in the definition of A190730. - _Petros Hadjicostas_, Feb 15 2021
%H A087322 Paolo Xausa, <a href="/A087322/b087322.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of the triangle, flattened)
%F A087322 T(n, k) = (n + 1)*2^k - 1 for n >= 1 and 1 <= k <= n.
%F A087322 From _Petros Hadjicostas_, Feb 15 2021: (Start)
%F A087322 Sum_{k=1..n} T(n,k) = A190730(n).
%F A087322 T(n,2) = 4*n + 3 for n >= 2.
%F A087322 T(n,n) = A087323(n).
%F A087322 T(n,n-1) = A099035(n) = (n+1)*2^(n-1) - 1 for n >= 2.
%F A087322 Recurrence: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) for n >= 2 and 2 <= k <= n with initial conditions the values of T(n, 1) and T(n,2).
%F A087322 Bivariate o.g.f.: Sum_{n,k>=1} T(n,k)*x^n*y^k = (4*x^3*y^2 - 2*x^2*y - 4*x*y - x + 3)*x*y/((1 - 2*x*y)^2*(1 - x*y)*(1 - x)^2). (End)
%e A087322 Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
%e A087322    3;
%e A087322    5, 11;
%e A087322    7, 15, 31;
%e A087322    9, 19, 39,  79;
%e A087322   11, 23, 47,  95, 191;
%e A087322   13, 27, 55, 111, 223, 447;
%e A087322   15, 31, 63, 127, 255, 511, 1023;
%e A087322   17, 35, 71, 143, 287, 575, 1151, 2303;
%e A087322   19, 39, 79, 159, 319, 639, 1279, 2559, 5119;
%e A087322   ...
%t A087322 A087322row[n_]:=NestList[2#+1&,2n+1,n-1];Array[A087322row,10] (* _Paolo Xausa_, Oct 17 2023 *)
%Y A087322 Cf. A087323, A099035, A190730.
%K A087322 nonn,tabl,easy
%O A087322 1,1
%A A087322 _Amarnath Murthy_, Sep 03 2003
%E A087322 Edited and extended by _David Wasserman_, May 06 2005
%E A087322 Name edited by _Petros Hadjicostas_, Feb 15 2021