A112851 a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.

0, 0, 3, 21, 81, 231, 546, 1134, 2142, 3762, 6237, 9867, 15015, 22113, 31668, 44268, 60588, 81396, 107559, 140049, 179949, 228459, 286902, 356730, 439530, 537030, 651105, 783783, 937251, 1113861, 1316136, 1546776, 1808664, 2104872, 2438667, 2813517, 3233097

Offset

0,3

Comments

A112851 is the fourth sequence in A112852.

Also the Wiener index of the (n-1)-triangular grid graph (indexed so the 0-triangular grid graph is the singleton). - Eric W. Weisstein, Sep 08 2017

References

L. B. W. Jolley, Summation of Series, Dover. N.Y., 1961, eq. (54), page 10.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Triangular Grid Graph.

Eric Weisstein's World of Mathematics, Wiener Index.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

4*a(n+1) = 1*2^2*3 + 2*3^2*4 + 3*4^2*5 + ... (n terms). [Jolley]

a(n) = Sum_{i=0..n} A000217(i-1)*A000217(i), where A000217(-1)=0. - Bruno Berselli, Feb 05 2014

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>5. - Harvey P. Dale, Oct 28 2014

G.f.: 3*x^2*(1 + x) / (1 - x)^6. - Colin Barker, Sep 08 2017

a(n) = (1/2) * Sum_{k=0..n} C(k^2,2). - Wesley Ivan Hurt, Sep 23 2017

a(n) = Sum_{i=0..n} A000217(i)*A033428(n-i). - Bruno Berselli, Mar 06 2018

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=2} 1/a(n) = 40*(16*log(2) - 11)/9.

Sum_{n>=2} (-1)^n/a(n) = 20*(8*Pi - 25)/9. (End)

Maple

a:=n->sum(j^4-j^2, j=0..n)/4: seq(a(n), n=0..36); # Zerinvary Lajos, May 08 2008

Mathematica

Table[(n - 1) n (n + 1)(n + 2)(2 n + 1)/40, {n, 0, 30}] (* Josh Locker *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 3, 21, 81, 231}, 40] (* Harvey P. Dale, Oct 28 2014 *)

Prog

(Magma) [(n-1)*n*(n+1)*(n+2)*(2*n+1)/40: n in [0..40]]; // Vincenzo Librandi, Feb 06 2014
(PARI) for(n=0, 50, print1((n-1)*n*(n+1)*(n+2)*(2*n+1)/40, ", ")) \\ G. C. Greubel, Jul 23 2017
(PARI) concat(vector(2), Vec(3*x^2*(1 + x) / (1 - x)^6 + O(x^30))) \\ Colin Barker, Sep 08 2017

Crossrefs

Partial sums of sequence A006011.

Cf. A000217, A033428, A112851, A112852.

Sequence in context: A110450 A176646 A102832 * A253943 A034490 A071351

Adjacent sequences: A112848 A112849 A112850 * A112852 A112853 A112854

Keyword

nonn,easy

Author

Alford Arnold, Sep 24 2005

Extensions

More terms from Josh Locker (jlocker(AT)mail.rochester.edu) and Michael W. Motily (mwm5036(AT)psu.edu), Oct 04 2005

Status

approved