A034263 a(n) = binomial(n+4,4)*(4*n+5)/5.

1, 9, 39, 119, 294, 630, 1218, 2178, 3663, 5863, 9009, 13377, 19292, 27132, 37332, 50388, 66861, 87381, 112651, 143451, 180642, 225170, 278070, 340470, 413595, 498771, 597429, 711109, 841464, 990264, 1159400, 1350888, 1566873, 1809633, 2081583, 2385279

Offset: 0

Author

Clark Kimberling, Barry E. Williams, Dec 13 1999

Keywords

• nonn
• easy

Comments

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005

5-dimensional form of hexagonal-based pyramid numbers. - Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005

Convolution of triangular numbers (A000217) and hexagonal numbers (A000384). - Bruno Berselli, Jun 27 2013

Examples

			By the third comment: A000217(1..6) and A000384(1..6) give the term a(5) = 1*21+5*15+12*10+22*6+35*3+51*1 = 630. - _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.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/4).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
• Index to sequences related to pyramidal numbers.

Crossrefs

Partial sums of A002417.

Cf. A000217, A000384, A000292, A034261, A059599, A093561.

Cf. similar sequences listed in A254142.

Programs

GAP

List([0..35], n-> (4*n+5)*Binomial(n+4,4)/5); # G. C. Greubel, Aug 28 2019

Magma

[(4*n+5)*Binomial(n+4,4)/5: n in [0..35]]; // G. C. Greubel, Aug 28 2019

Maple

a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120: seq(a(n),n=0..35); # Emeric Deutsch, Nov 18 2005

Mathematica

Table[Binomial[n+4, 4]*(4*n+5)/5, {n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
a[n_] := (1+n)(2+n)(3+n)(4+n)(4n+5)/120; Array[a, 36, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 39, 119, 294, 630}, 36] (* or *)
CoefficientList[ Series[(1+3*x)/(1-x)^6, {x, 0, 35}], x] (* Robert G. Wilson v, Feb 26 2015 *)
Table[Sum[-x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 1, 30}]/4 (* Horst H. Manninger, Jun 19 2025 *)

PARI

a(n)=(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120 \\ Charles R Greathouse IV, Sep 24 2015, corrected by Altug Alkan, Aug 15 2017

Sage

[(4*n+5)*binomial(n+4,4)/5 for n in (0..35)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = A093561(n+5, 5).

a(n) = A034261(n+1, 3).

G.f.: (1+3*x)/(1-x)^6.

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120. - Emeric Deutsch and Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005, corrected by Eric Rowland, Aug 15 2017

a(-n-4) = -A059599(n). - Bruno Berselli, Aug 23 2011

a(n) = Sum_{i=1..n+1} i*A000292(i). - Bruno Berselli, Jan 23 2015

Sum_{n>=0} 1/a(n) = 28300/231 - 1280*Pi/77 - 7680*log(2)/77. - Amiram Eldar, Feb 15 2022

Extensions

Corrected and extended by N. J. A. Sloane, Apr 21 2000