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A112851
a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.
3
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
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
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.
Sequence in context: A110450 A176646 A102832 * A253943 A034490 A071351
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