A134481 Row sums of triangle A134480.

1, 5, 20, 50, 100, 175, 280, 420, 600, 825, 1100, 1430, 1820, 2275, 2800, 3400, 4080, 4845, 5700, 6650, 7700, 8855, 10120, 11500, 13000, 14625, 16380, 18270, 20300, 22475, 24800, 27280, 29920, 32725, 35700, 38850, 42180, 45695

Offset: 0

Author

Gary W. Adamson, Oct 27 2007

Keywords

• nonn
• easy

Comments

Binomial transform of [1, 4, 11, 4, 1, -1, 1, -1, 1, -1, ...]. Also, 1 followed by 5 * A000292, the tetrahedral numbers; i.e., 1, then 5 * (1, 4, 10, 25, 35, ...).

If Y is a 5-subset of an n-set X then, for n>=8, a(n-7) is the number of 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Examples

			a(2) = 20 = sum of row 3 terms of triangle A134480: (9 + 7 + 4).
a(3) = 50 = (1, 3, 3, 1) dot (1, 4, 11, 4) = (1 + 12 + 33 + 4).
a(2) = 20 = 2*1 + 3*2 + 4*3; a(5) = 5*1 + 6*2 + 7*3 + 8*4 + 9*5 + 10*6. - _Bruno Berselli_, Dec 16 2013
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A134480, A000292.

Programs

Magma

I:=[1, 5, 20, 50, 100]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 29 2012

Mathematica

CoefficientList[Series[1+5x/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 29 2012 *)

Formula

a(n) = 5*binomial(n+2,3) for n>0. - Milan Janjic, Dec 28 2007

G.f.: 1 + 5*x / (1-x)^4. - R. J. Mathar, Apr 04 2012

a(n) = Sum_{i=0..n} (n+i)*(1+i) for n > 0. - Bruno Berselli, Dec 16 2013

E.g.f.: 1 + 5*exp(x)*x*(6 + 6*x + x^2)/6. - Stefano Spezia, Oct 09 2023