A132117 Binomial transform of [1, 7, 17, 17, 6, 0, 0, 0, ...].

1, 8, 32, 90, 205, 406, 728, 1212, 1905, 2860, 4136, 5798, 7917, 10570, 13840, 17816, 22593, 28272, 34960, 42770, 51821, 62238, 74152, 87700, 103025, 120276, 139608, 161182, 185165, 211730, 241056, 273328, 308737, 347480, 389760, 435786, 485773, 539942, 598520

Offset: 1

Author

Gary W. Adamson, Aug 10 2007

Keywords

• nonn
• easy

Comments

Equals row sums of triangle A178067. - Gary W. Adamson, May 18 2010

Antidiagonal sums of the convolution array A213771. - Clark Kimberling, Jul 04 2012

Partial sums of A081436. - J. M. Bergot, Jun 20 2013

Examples

			a(3) = 32 = (1, 2, 1) dot (1, 7, 17) = (1 + 14 + 17).
a(5) = 15^2 - (10+6+3+1) = A000537(5) - A000292(4) = 225 - 20 = 205. - _Bruno Berselli_, May 01 2010
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A178067. - Gary W. Adamson, May 18 2010

Programs

Maple

a:= n-> (Matrix([[0,0,2,13,46]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [5,-10,10,-5,1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..29); # Alois P. Heinz, Aug 07 2008
a:= n-> (4+(6+(8+6*n)*n)*n)*n/24: seq(a(n),n=1..40); # Alois P. Heinz, Aug 07 2008

Mathematica

Table[(4 n + 6 n^2 + 8 n^3 + 6 n^4) / 24, {n, 50}] (* Vincenzo Librandi, Jun 21 2013 *)

PARI

a(n) = (4*n+6*n^2+8*n^3+6*n^4)/24 \\ Charles R Greathouse IV, Sep 03 2011

Formula

Let M = the infinite lower triangular matrix of the natural numbers: [1; 2,3; 4,5,6; ...]; and V = [1, 2, 3, ...]. Then M*V = A132117.

O.g.f.: -x(1+x)(2x+1)/(-1+x)^5. - R. J. Mathar, Apr 02 2008

a(n) = (4*n + 6*n^2 + 8*n^3 + 6*n^4)/24. - Alois P. Heinz, Aug 07 2008

a(n) = A000217(n)^2 - Sum_{i=1..n-1} A000217(i) = n*(n+1)*(3*n^2+n+2)/12. - Bruno Berselli, May 01 2010

Extensions

More terms from R. J. Mathar, Apr 02 2008