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

0,3

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 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)/(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

Partial sums of A001296.

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

Cf. A000217, A000326, A213548.

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