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

0,2

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

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4). - R. J. Mathar, Oct 30 2010

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)

Example

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).

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 *)

Crossrefs

Cf. A002064, A113127.

Sequence in context: A357286 A277974 A089092 * A049611 A084851 A094706

Adjacent sequences: A181524 A181525 A181526 * A181528 A181529 A181530

Keyword

nonn,easy

Author

Gary W. Adamson, Oct 26 2010

Status

approved