A132124 a(n) = n*(n+1)*(8*n + 1)/6.

0, 3, 17, 50, 110, 205, 343, 532, 780, 1095, 1485, 1958, 2522, 3185, 3955, 4840, 5848, 6987, 8265, 9690, 11270, 13013, 14927, 17020, 19300, 21775, 24453, 27342, 30450, 33785, 37355, 41168, 45232, 49555, 54145, 59010, 64158, 69597, 75335, 81380, 87740, 94423

Offset: 0

Author

Reinhard Zumkeller, Aug 12 2007

Keywords

• nonn
• easy

Comments

Convolution of the sequences (0,3,5,0,0,0,...) and (binomial(n+3, 3)), n >= 0. - Emeric Deutsch, Aug 30 2007

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000330, A033994, A132112, A050409.

Cf. A014105, A016061; A002412, A006331; A007585, A002378.

Row sums of A069011.

Programs

Maple

seq((1/6)*n*(n+1)*(8*n+1),n=0..40); # Emeric Deutsch, Aug 30 2007

Mathematica

a[n_] := n*(n + 1)*(8*n + 1)/6; Array[a, 42, 0] (* Amiram Eldar, May 20 2023 *)

Formula

a(n) = A132121(n,2) for n > 1.

G.f.: x*(3+5*x)/(1-x)^4. - Emeric Deutsch, Aug 30 2007

From Bruno Berselli, Nov 25 2010: (Start)

a(n) = n*A014105(n) - A016061(n-1), since A016061(n-1) = Sum_{k=0..n-1} A014105(k) (n > 0).

Also a(n) = A002412(n) + A006331(n) = A007585(n) + A002378(n). (End)

Sum_{n>=1} 1/a(n) = 54 - 24*(sqrt(2)+1)*Pi/7 - 24*(sqrt(2)+8)*log(2)/7 + 48*sqrt(2)*log(2-sqrt(2))/7. - Amiram Eldar, May 20 2023

E.g.f.: exp(x)*x*(18 + 33*x + 8*x^2)/6. - Stefano Spezia, Feb 21 2024