A101097 a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(2 + 4*n + n^2)/840.

1, 12, 69, 272, 846, 2232, 5214, 11088, 21879, 40612, 71643, 121056, 197132, 310896, 476748, 713184, 1043613, 1497276, 2110273, 2926704, 3999930, 5393960, 7184970, 9462960, 12333555, 15919956, 20365047, 25833664, 32515032, 40625376, 50410712

Offset: 1

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Keywords

• nonn
• easy

Comments

Fourth partial sums of cubes (A000578). Partial sums of A101094.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• 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 p. 15.
• C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link]
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A000217, A101102, A101094, A024166, A000537.

Programs

Magma

A000217:=func; [&+[A000217(k)^2*A000217(n-k+1): k in [1..n]]: n in [1..40]]; // Bruno Berselli, Sep 04 2013

Mathematica

Table[Binomial[n+4,5]*(2+4*n+n^2)/7, {n,0,50}] (* G. C. Greubel, Feb 17 2017 *)

PARI

{A101097(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(2+4*n+n^2)/840} \\ R. J. Mathar, Dec 06 2011

Sage

[binomial(n+4,5)*(2+4*n+n^2)/7 for n in (1..40)] # G. C. Greubel, Dec 01 2018

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) for k=4. - Alexander R. Povolotsky, May 17 2008

G.f.: x*(1 + 4*x + x^2)/(1-x)^8. - R. J. Mathar, Jun 13 2008

a(n) = Sum_{k=1..n} A000217(k)^2*A000217(n-k+1). - Bruno Berselli, Sep 04 2013

E.g.f.: x*(840 + 4200*x + 5040*x^2 + 2240*x^3 + 427*x^4 + 35*x^5 + x^6) *exp(x)/840. - G. C. Greubel, Dec 01 2018

Extensions

Edited by Ralf Stephan, Dec 16 2004