

A051836


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


14



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

5dimensional version of pentagonalbased pyramidal numbers.  Ben Creech (mathroxmysox(AT)yahoo.com)
If Y is a 3subset of an nset X then, for n>=7, a(n6) is the number of 7subsets 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)Andrasfai graph.  Eric W. Weisstein, Apr 14 2017


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 18.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Andrasfai 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).


FORMULA

a(n) = C(n+4, n)*(3n+5)/5.
G.f.: x*(1+2*x)/(1x)^6. (adapted by Vincenzo Librandi, Jul 04 2017)


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[n1] 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


CROSSREFS

Partial sums of A001296.
Cf. A093560 ((3, 1) Pascal, column m=5).
Sequence in context: A316148 A014820 A070736 * A278670 A301771 A070051
Adjacent sequences: A051833 A051834 A051835 * A051837 A051838 A051839


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



