OFFSET
0,3
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
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 entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
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
EXAMPLE
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
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 *)
PROG
(PARI) a(n)=n*(n+1)*(n+2)*(n+3)*(3*n+2)/120 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [0] cat [Binomial(n+4, n)*(3*n+5)/5: n in [0..40]]; // Vincenzo Librandi, Jul 04 2017
(SageMath) [((3*n+2)/(n+4))*binomial(n+4, 5) for n in range(41)] # G. C. Greubel, Dec 27 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 12 1999
EXTENSIONS
Simpler definition from Ben Creech (mathroxmysox(AT)yahoo.com), Nov 13 2005
STATUS
approved