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, 10115952, 12059817, 14305347, 16888872

Offset

0,2

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

Albert 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).

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 216.

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).

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

E.g.f.: exp(x)*(720 + 7200*x + 12600*x^2 + 7200*x^3 + 1650*x^4 + 156*x^5 + 5*x^6)/720. - Stefano Spezia, Oct 30 2025

Sum_{n>=0} 1/a(n) = 59195/266 - (3125/133)*(cot(Pi/5)*Pi + sqrt(5)*log(phi) + 5*log(5)/2), where phi is the golden ratio (A001622). - Amiram Eldar, Nov 17 2025

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, Jun 18 2005

Mathematica

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

Prog

(Magma) [(5*n+6)*Binomial(n+5, 5)/6: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
(PARI) vector(40, n, (5*n+1)*binomial(n+4, 5)/6) \\ G. C. Greubel, Aug 28 2019
(SageMath) [(5*n+6)*binomial(n+5, 5)/6 for n in (0..40)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..40], n-> (5*n+6)*Binomial(n+5, 5)/6); # G. C. Greubel, Aug 28 2019

Crossrefs

Partial sums of A027800.

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

Cf. A001622, A143130.

Cf. similar sequences listed in A254142.

Sequence in context: A224154 A079547 A034264 * A224405 A201150 A114030

Adjacent sequences: A051943 A051944 A051945 * A051947 A051948 A051949

Keyword

nonn,easy

Author

Barry E. Williams, Dec 20 1999

Extensions

Corrected and extended by Emeric Deutsch, Jun 18 2005

a(34) corrected by Sean A. Irvine, Nov 02 2025

Status

approved