A203552 a(n) = n*(5*n^2 - 3*n + 4) / 6.

0, 1, 6, 20, 48, 95, 166, 266, 400, 573, 790, 1056, 1376, 1755, 2198, 2710, 3296, 3961, 4710, 5548, 6480, 7511, 8646, 9890, 11248, 12725, 14326, 16056, 17920, 19923, 22070, 24366, 26816, 29425, 32198, 35140, 38256, 41551, 45030, 48698

Offset: 0

Author

Michael Somos, Jan 02 2012

Keywords

• nonn
• easy

Examples

			G.f. = x + 6*x^2 + 20*x^3 + 48*x^4 + 95*x^5 + 166*x^6 + 266*x^7 + 400*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See p. 11.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000125, A134238, A203551.

Programs

Magma

I:=[0, 1, 6, 20]; [n le 4 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, Jan 07 2012

Mathematica

LinearRecurrence[{4,-6,4,-1},{0,1,6,20},40] (* Vincenzo Librandi, Jan 07 2012 *)

PARI

{a(n) = n * (5*n^2 - 3*n + 4) / 6};

Formula

a(n) = Sum_{k = 1..n} A(k-1, n-k) where A(i, j) = i^2 + i*j + j^2 + i + j + 1.

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

a( n) = -A203551(-n) for all n in Z.

a(n)-a(n-1) = A134238(n). - Bruno Berselli, Jan 03 2012

a(n) = 4*A000125(n) + 2*A000125(n+1) - A000125(n+3). - Ivan N. Ianakiev, Aug 21 2013

E.g.f.: x*(5*x^2 + 12*x + 6)*exp(x)/6. - G. C. Greubel, Aug 12 2018