A051836 a(n) = n*(n+1)*(n+2)*(n+3)*(3*n+2)/120.

0, 1, 8, 33, 98, 238, 504, 966, 1716, 2871, 4576, 7007, 10374, 14924, 20944, 28764, 38760, 51357, 67032, 86317, 109802, 138138, 172040, 212290, 259740, 315315, 380016, 454923, 541198, 640088, 752928, 881144, 1026256, 1189881, 1373736, 1579641, 1809522, 2065414

Offset: 0

Author

Barry E. Williams, Dec 12 1999

Keywords

• nonn
• easy

Comments

5-dimensional version of pentagonal-based pyramidal numbers. - Ben Creech (mathroxmysox(AT)yahoo.com)

If Y is a 3-subset of an n-set X then, for n>=7, a(n-6) is the number of 7-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007

Antidiagonal sums of the convolution array A213548. - Clark Kimberling, Jun 17 2012

After 0, convolution of nonzero triangular numbers (A000217) and nonzero pentagonal numbers (A000326). - Bruno Berselli, Jun 27 2013

a(n) is also the number of odd chordless cycles in the graph complement of the (n+1)-Andrásfai graph. - Eric W. Weisstein, Apr 14 2017

Examples

			By the fourth comment: A000217(1..6) and A000326(1..6) give the term a(6) = 1*21+5*15+12*10+22*6+35*3+51*1 = 504. - _Bruno Berselli_, Jun 27 2013
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Andrásfai Graph.
• Eric Weisstein's World of Mathematics, Chordless Cycle.
• Eric Weisstein's World of Mathematics, Graph Complement.
• Index to sequences related to pyramidal numbers.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Partial sums of A001296.

Cf. A093560 ((3, 1) Pascal, column m=5).

Cf. A000217, A000326, A213548.

Programs

Magma

[0] cat [Binomial(n+4, n)*(3*n+5)/5: n in [0..40]]; // Vincenzo Librandi, Jul 04 2017

Maple

with (combinat):a[0]:=0:for n from 1 to 50 do a[n]:=stirling2(n+2,n)+a[n-1] od: seq(a[n], n=0..34); # Zerinvary Lajos, Mar 17 2008

Mathematica

Table[n(n + 1)(n + 2)(n + 3)(3n + 2)/120, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
CoefficientList[Series[x (1 + 2 x) / (1 - x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Jul 04 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,8,33,98,238},40] (* Harvey P. Dale, Jun 01 2018 *)

PARI

a(n)=n*(n+1)*(n+2)*(n+3)*(3*n+2)/120 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[((3*n+2)/(n+4))*binomial(n+4,5) for n in range(41)] # G. C. Greubel, Dec 27 2023

Formula

a(n) = C(n+4, n)*(3n+5)/5.

G.f.: x*(1+2*x)/(1-x)^6. (adapted by Vincenzo Librandi, Jul 04 2017)

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 135*sqrt(3)*Pi/14 - 1215*log(3)/14 + 925/21.

Sum_{n>=1} (-1)^(n+1)/a(n) = 135*sqrt(3)*Pi/7 - 880*log(2)/7 - 355/21. (End)

E.g.f.: (1/5!)*x*(120 + 360*x + 240*x^2 + 50*x^3 + 3*x^4)*exp(x). - G. C. Greubel, Dec 27 2023

Extensions

Simpler definition from Ben Creech (mathroxmysox(AT)yahoo.com), Nov 13 2005