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

0,2

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

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.

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

Example

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

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 *)

Prog

(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
(Magma) [(4*n+5)*Binomial(n+4, 4)/5: n in [0..35]]; // G. C. Greubel, Aug 28 2019
(Sage) [(4*n+5)*binomial(n+4, 4)/5 for n in (0..35)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..35], n-> (4*n+5)*Binomial(n+4, 4)/5); # G. C. Greubel, Aug 28 2019

Crossrefs

Partial sums of A002417.

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

Cf. similar sequences listed in A254142.

Sequence in context: A023163 A054121 A139594 * A060929 A212143 A294845

Adjacent sequences: A034260 A034261 A034262 * A034264 A034265 A034266

Keyword

nonn,easy

Author

Clark Kimberling, Barry E. Williams, Dec 13 1999

Extensions

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

Status

approved