A027800 a(n) = (n+1)*binomial(n+4, 4).

1, 10, 45, 140, 350, 756, 1470, 2640, 4455, 7150, 11011, 16380, 23660, 33320, 45900, 62016, 82365, 107730, 138985, 177100, 223146, 278300, 343850, 421200, 511875, 617526, 739935, 881020, 1042840, 1227600, 1437656, 1675520, 1943865, 2245530, 2583525

Offset: 0

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 9-subsequences of [1, n] with just 4 contiguous pairs.

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 19 2005

Equals binomial transform of [1, 9, 26, 34, 21, 5, 0, 0, 0, ...]. - Gary W. Adamson, Jul 27 2008

a(n) equals the coefficient of x^4 of the characteristic polynomial of the (n+4) X (n+4) matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 08 2011

Convolution of triangular numbers (A000217) and heptagonal numbers (A000566). - Bruno Berselli, Jun 27 2013

Examples

			By the fifth comment: A000217(1..6) and A000566(1..6) give the term a(6) = 1*21 + 7*15 + 18*10 + 34*6 + 55*3 + 81*1 = 756. - _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 (p.233, # 9).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Mina Aganagic, Albrecht Klemm and Cumrun Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.
• 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 A002418.

Cf. A000332, A093562 ((5, 1) Pascal, column m=5).

Programs

GAP

List([0..40], n-> (n+1)*Binomial(n+4,4)); # G. C. Greubel, Aug 28 2019

Magma

[(n+1)*Binomial(n+4,4): n in [0..40]]; // G. C. Greubel, Aug 28 2019

Maple

a:=n->(n+1)^2*(n+2)*(n+3)*(n+4)/24: seq(a(n),n=0..40); # Emeric Deutsch

Mathematica

Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n+4, n+4}], x], x^4], {n, 0, 40}] (* John M. Campbell, Jul 08 2011 *)
Table[(n+1)Binomial[n+4, 4], {n,0,40}] (* or *) CoefficientList[Series[ (1+4x)/(1-x)^6, {x,0,40}], x] (* Michael De Vlieger, Jul 14 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,45,140,350,756},40] (* Harvey P. Dale, Aug 04 2020 *)

PARI

vector(40, n, n*binomial(n+3,4)) \\ G. C. Greubel, Aug 28 2019

Sage

[(n+1)*binomial(n+4,4) for n in (0..40)] # G. C. Greubel, Aug 28 2019

Formula

G.f.: (1+4*x)/(1-x)^6.

a(n) = (n+1)*A000332(n+4).

Sum_{n>=0} 1/a(n) = (2/3)*Pi^2 - 49/9. - Jaume Oliver Lafont, Jul 14 2017

E.g.f.: exp(x)*(24 + 216*x + 312*x^2 + 136*x^3 + 21*x^4 + x^5)/24. - Stefano Spezia, May 08 2021

Sum_{n>=0} (-1)^n/a(n) = Pi^2/3 - 80*log(2)/3 + 145/9. - Amiram Eldar, Jan 28 2022