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%I #29 Feb 08 2022 22:26:28
%S 1,4,13,38,103,264,649,1546,3595,8204,18445,40974,90127,196624,426001,
%T 917522,1966099,4194324,8912917,18874390,39845911,83886104,176160793,
%U 369098778,771751963,1610612764,3355443229,6979321886,14495514655,30064771104,62277025825
%N Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).
%C A181527 = Partial sums of (A002064 Cullen numbers: n*2^n+1). - _Vladimir Joseph Stephan Orlovsky_, Jul 09 2011
%C Form a triangle with T(1,1) = n, T(2,1) = T(2,2) = n-1, T(3,1) = T(3,3) = n-2, ..., T(n,1) = T(n,n) = 1. The interior members are T(i,j) = T(i-1,j-1) + T(i-1,j). The sum of all members for a triangle of size n is a(n-1). Example for n = 5: row(1) = 5; row(2) = 4, 4; row(3) = 3, 8, 3; row(4) = 2, 11, 11, 2; row(5) = 1, 13, 22, 13, 1. The sum of all members is 103 = a(4). - _J. M. Bergot_, Oct 16 2012
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4). - _R. J. Mathar_, Oct 30 2010
%F Binomial transform of A113127; (1, 3, 7, 15, 31, ...) convolved with (1, 1, 3, 7, 15, 31, ...).
%F From _R. J. Mathar_, Oct 30 2010: (Start)
%F a(n) = 3+ n + 2^(n+1)*(n-1) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
%F G.f.: ( 1-2*x+2*x^2 ) / ( (2*x-1)^2*(x-1)^2 ). (End)
%e a(4) = 103 = (1, 1, 3, 7, 15) dot (31, 15, 7, 3, 1) = (31 + 15 + 21, + 21 + 15)
%e a(3) = 38 = (1, 3, 3, 1) dot (1, 3, 6, 10) = (1 + 9 + 18 + 10).
%t Accumulate[Table[n*2^n + 1, {n, 0, 60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 09 2011 *)
%t LinearRecurrence[{6,-13,12,-4},{1,4,13,38},40] (* _Harvey P. Dale_, Apr 14 2016 *)
%Y Cf. A002064, A113127.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Oct 26 2010
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