A181527 Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).

1, 4, 13, 38, 103, 264, 649, 1546, 3595, 8204, 18445, 40974, 90127, 196624, 426001, 917522, 1966099, 4194324, 8912917, 18874390, 39845911, 83886104, 176160793, 369098778, 771751963, 1610612764, 3355443229, 6979321886, 14495514655, 30064771104, 62277025825

Offset: 0

Author

Gary W. Adamson, Oct 26 2010

Keywords

• nonn
• easy

Comments

A181527 = Partial sums of (A002064 Cullen numbers: n*2^n+1). - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

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

Examples

			a(4) = 103 = (1, 1, 3, 7, 15) dot (31, 15, 7, 3, 1) = (31 + 15 + 21, + 21 + 15)
a(3) = 38 = (1, 3, 3, 1) dot (1, 3, 6, 10) = (1 + 9 + 18 + 10).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A002064, A113127.

Programs

Mathematica

Accumulate[Table[n*2^n + 1, {n, 0, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jul 09 2011 *)
LinearRecurrence[{6,-13,12,-4},{1,4,13,38},40] (* Harvey P. Dale, Apr 14 2016 *)

Formula

Binomial transform of A113127; (1, 3, 7, 15, 31, ...) convolved with (1, 1, 3, 7, 15, 31, ...).

From R. J. Mathar, Oct 30 2010: (Start)

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).

G.f.: ( 1-2*x+2*x^2 ) / ( (2*x-1)^2*(x-1)^2 ). (End)