A051946 Expansion of g.f.: (1+4*x)/(1-x)^7.

1, 11, 56, 196, 546, 1302, 2772, 5412, 9867, 17017, 28028, 44408, 68068, 101388, 147288, 209304, 291669, 399399, 538384, 715484, 938630, 1216930, 1560780, 1981980, 2493855, 3111381, 3851316, 4732336, 5775176, 7002776, 8440432

Offset: 0

Author

Barry E. Williams, Dec 20 1999

Keywords

• nonn
• easy

Comments

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

Equals row sums of triangle A143130, and binomial transform of {1, 10, 35, 60, 55, 26, 5, 0, 0, 0, ...}. - Gary W. Adamson, Jul 27 2008

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 5).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Partial sums of A027800.

Cf. A093562 ((5, 1) Pascal, column m=6).

Cf. A143130.

Cf. similar sequences listed in A254142.

Programs

GAP

List([0..40], n-> (5*n+6)*Binomial(n+5,5)/6); # G. C. Greubel, Aug 28 2019

Magma

[(5*n+6)*Binomial(n+5,5)/6: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

Maple

a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720: seq(a(n),n=0..35); # Emeric Deutsch

Mathematica

CoefficientList[Series[(1+4x)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

PARI

vector(40, n, (5*n+1)*binomial(n+4,5)/6) \\ G. C. Greubel, Aug 28 2019

Sage

[(5*n+6)*binomial(n+5,5)/6 for n in (0..40)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = binomial(n+5,5)*(5*n+6)/6.

a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720. - Emeric Deutsch, Jun 18 2005

a(n) = A034264(n+1). - R. J. Mathar, Oct 14 2008

Extensions

Corrected and extended by Emeric Deutsch, Jun 18 2005