A081498 Consider the triangle in which the n-th row starts with n, contains n terms and the difference of successive terms is 1,2,3,... up to n-1. Sequence gives row sums.

1, 3, 5, 6, 5, 1, -7, -20, -39, -65, -99, -142, -195, -259, -335, -424, -527, -645, -779, -930, -1099, -1287, -1495, -1724, -1975, -2249, -2547, -2870, -3219, -3595, -3999, -4432, -4895, -5389, -5915, -6474, -7067, -7695, -8359, -9060, -9799, -10577, -11395, -12254, -13155, -14099, -15087, -16120

Offset: 1

Author

Amarnath Murthy, Mar 25 2003

Keywords

• sign
• easy

Comments

The triangle whose row sums are being considered is:

1;

2, 1;

3, 2, 0;

4, 3, 1, -2;

5, 4, 2, -1, -5;

6, 5, 3, 0, -4, -9;

7, 6, 4, 1, -3, -8, -14;

The leading diagonal is given by A080956(n-1) = n*(3-n)/2.

Examples

			G.f. = x * (1 + 3*x + 5*x^2 + 6*x^3 + 5*x^4 + x^5 - 7*x^6 - 20*x^7 - 39*x^8 - 65*x^9 + ...).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A008778, A080956, A081499.

Programs

GAP

List([1..50],n->n^2-Binomial(n+1,n-2)); # Muniru A Asiru, Mar 05 2019

Magma

[n*(1+6*n-n^2)/6: n in [1..50]]; // G. C. Greubel, Mar 06 2019

Maple

seq(n^2-binomial(n+1,n-2),n=1..50); # C. Ronaldo
[seq(binomial(n,2)+binomial(n,1)-binomial(n,3), n=1..49)]; # Zerinvary Lajos, Jul 23 2006

Mathematica

LinearRecurrence[{4,-6,4,-1}, {1,3,5,6}, 50] (* G. C. Greubel, Mar 06 2019 *)

PARI

{a(n) = if( n< 0, n = -2 - n; polcoeff( (1 + x - x^2) / (1 - x)^4 + x * O(x^n), n), polcoeff( (1 - x - x^2) / (1 - x)^4 + x * O(x^n), n))} /* Michael Somos, Jul 04 2012 */

PARI

vector(50, n, n*(1+6*n-n^2)/6) \\ G. C. Greubel, Mar 06 2019

Sage

[n*(1+6*n-n^2)/6 for n in (1..50)] # G. C. Greubel, Mar 06 2019

Formula

a(n) = n^2 - binomial(n+1, n-2). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

a(n) = binomial(n,2)+binomial(n,1)-binomial(n,3). - Zerinvary Lajos, Jul 23 2006

a(n) = n*(1+6*n-n^2)/6. - Karen A. Yeats, Nov 20 2006

From Michael Somos, Jul 04 2012: (Start)

G.f.: x * (1 - x - x^2) / (1 - x)^4.

a(-1 - n) = A008778(n). (End)

E.g.f.: x*(6 +3*x -x^2)*exp(x)/6. - G. C. Greubel, Mar 06 2019

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004

Offset changed to 1 at the suggestion of Michel Marcus, Mar 05 2019

Formulas and programs addapted for offset 1 by Michel Marcus, Mar 05 2019