A086020 a(n) = Sum_(i=1..n) binomial(i+2,3)^2 [ Sequential sums of the tetragonal numbers or "tetras" (pyramidal, square) raised to power 2 (drawn from the 4th diagonal - left or right - of Pascal's Triangle) ].

1, 17, 117, 517, 1742, 4878, 11934, 26334, 53559, 101959, 183755, 316251, 523276, 836876, 1299276, 1965132, 2904093, 4203693, 5972593, 8344193, 11480634, 15577210, 20867210, 27627210, 36182835, 46915011, 60266727, 76750327

Offset

1,2

Comments

Kekulé numbers for certain benzenoids (see the Cyvin-Gutman reference, p. 243; expression in (13.26) yields same sequence with offset 0). - Emeric Deutsch, Aug 02 2005

Partial sums of A001249. - R. J. Mathar, Aug 19 2008

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988.

John Engbers and Christopher Stocker, Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients, Integers 16 (2016), #A58.

Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = Sum_(i=1..n) binomial(i+2, 3)^2.

a(n) = ( C(n+3, 4)/35 )*( 35 + 84*C(n-1, 1) + 70*C(n-1, 2) + 20*C(n-1, 3) ).

a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+3)(5*n^2 + 15*n + 1)/2520. - Emeric Deutsch, Aug 02 2005

O.g.f: x*(1+x)*(1 + 8*x + x^2)/(1-x)^8. - R. J. Mathar, Aug 19 2008

Example

a(8) = Sum_{i=1..8} binomial(i+2,3)^2 = (20*(8^7) + 210*(8^6) + 854*(8^5) + 1680*(8^4) + 1610*(8^3) + 630*(8^2) + 36*8)/7! = 26334.

Maple

a:=n->n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: seq(a(n), n=1..31); # Emeric Deutsch

Mathematica

Accumulate[Binomial[Range[30]+2, 3]^2]  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 17, 117, 517, 1742, 4878, 11934, 26334}, 30] (* Harvey P. Dale, Aug 17 2014 *)

Prog

(PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520 \\ Charles R Greathouse IV, May 18 2015
(Magma) [n*(n+1)*(n+2)*(n+3)*(2*n+3)*(5*n^2+15*n+1)/2520: n in [1..30]]; // G. C. Greubel, Nov 22 2017

Crossrefs

Cf. A000292, A087127, A024166, A085438, A085439, A085440, A085441, A085442, A000332, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.

Sequence in context: A293877 A044349 A044730 * A056117 A196575 A003109

Adjacent sequences: A086017 A086018 A086019 * A086021 A086022 A086023

Keyword

easy,nice,nonn

Author

André F. Labossière, Jul 17 2003

Status

approved