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A027800 a(n) = (n+1)*binomial(n+4, 4). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

REFERENCES

A. 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

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

FORMULA

G.f.: (1+4x)/(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

EXAMPLE

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

MAPLE

a:=n->(n+1)^2*(n+2)*(n+3)*(n+4)/24: seq(a(n), n=0..36); # 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, 31}] (* or *)

CoefficientList[Series[(1 + 4 x)/(1 - x)^6, {x, 0, 31}], x] (* Michael De Vlieger, Jul 14 2017 *)

CROSSREFS

Partial sums of A002418.

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

Sequence in context: A179095 A213188 A037270 * A005714 A175705 A143671

Adjacent sequences:  A027797 A027798 A027799 * A027801 A027802 A027803

KEYWORD

nonn,easy

AUTHOR

thi ngoc dinh (via R. K. Guy)

STATUS

approved

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Last modified February 19 11:04 EST 2018. Contains 299330 sequences. (Running on oeis4.)