A067056 a(n) = (1)*(2 + 3 + 4 + ... + n) + (1 + 2)*(3 + 4 + 5 + ... + n) + (1 + 2 + 3)*(4 + 5 + 6 + ... + n) + ... + (1 + 2 + 3 + ... + n-1)*n.

1, 2, 14, 54, 154, 364, 756, 1428, 2508, 4158, 6578, 10010, 14742, 21112, 29512, 40392, 54264, 71706, 93366, 119966, 152306, 191268, 237820, 293020, 358020, 434070, 522522, 624834, 742574, 877424, 1031184, 1205776, 1403248, 1625778, 1875678, 2155398, 2467530

Offset: 1

Author

Amarnath Murthy, Jan 02 2002

Keywords

• nonn
• easy

Examples

			a(4) = (1)*(2+3+4) + (1+2)*(3+4) + (1+2+3)*(4) = 9 + 21 + 24 = 54.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Bünyamin Şahin, Level Polynomials of Rooted Trees, 2023.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000217, A001105, A005585.

Programs

Mathematica

Join[{1},Table[Total[Total[#[[1]]Total[#[[2]]]]&/@Table[TakeDrop[ Range[ k],n],{n,k-1}]],{k,2,40}]] (* Requires Mathematica version 10 or later *)  (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,14,54,154,364,756},40] (* Harvey P. Dale, Jul 17 2020 *)

PARI

t(n) = n*(n+1)/2;
a(n) = if (n=1, 1, sum(k=1, n-1, t(k)*(t(n) - t(k)))); \\ Michel Marcus, Mar 06 2018

PARI

Vec(x*(1 - 4*x + 17*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6) / (1 - x)^6 + O(x^60)) \\ Colin Barker, Mar 06 2018

Formula

a(n) = Sum_{r=1..n-1} t(r)*(t(n) - t(r)), where t(r) is the r-th triangular number, n>1.

a(n) = n*(2*n^4 + 5*n^3 - 5*n - 2)/60 = (n-1)*n*(n+1)*(n+2)*(2*n+1)/60, n>1. - Ralf Stephan, Apr 30 2004

a(n) = 2*A005585(n-1), n>1. - R. J. Mathar, May 20 2013

a(n) = Sum_{i=1..n} A000217(i)*A001105(n-i) for n>1, a(1)=1. - Bruno Berselli, Mar 06 2018

From Colin Barker, Mar 06 2018: (Start)

G.f.: x*(1 - 4*x + 17*x^2 - 20*x^3 + 15*x^4 - 6*x^5 + x^6) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>7.

(End)

Extensions

More terms from Jason Earls, Jan 11 2002