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a(n) = (n+1)*binomial(n+4, 4).
9

%I #62 Jan 28 2022 04:30:21

%S 1,10,45,140,350,756,1470,2640,4455,7150,11011,16380,23660,33320,

%T 45900,62016,82365,107730,138985,177100,223146,278300,343850,421200,

%U 511875,617526,739935,881020,1042840,1227600,1437656,1675520,1943865,2245530,2583525

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

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

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 19 2005

%C Equals binomial transform of [1, 9, 26, 34, 21, 5, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 27 2008

%C 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

%C Convolution of triangular numbers (A000217) and heptagonal numbers (A000566). - _Bruno Berselli_, Jun 27 2013

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%D Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 9).

%H Vincenzo Librandi, <a href="/A027800/b027800.txt">Table of n, a(n) for n = 0..1000</a>

%H Mina Aganagic, Albrecht Klemm and Cumrun Vafa, <a href="http://arXiv.org/abs/hep-th/0105045">Disk Instantons, Mirror Symmetry and the Duality Web</a>, arXiv:hep-th/0105045, 2001.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.

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

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

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

%F 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

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

%e 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

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

%t Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n+4, n+4}], x], x^4], {n, 0, 40}] (* _John M. Campbell_, Jul 08 2011 *)

%t 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 *)

%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,45,140,350,756},40] (* _Harvey P. Dale_, Aug 04 2020 *)

%o (PARI) vector(40, n, n*binomial(n+3,4)) \\ _G. C. Greubel_, Aug 28 2019

%o (Magma) [(n+1)*Binomial(n+4,4): n in [0..40]]; // _G. C. Greubel_, Aug 28 2019

%o (Sage) [(n+1)*binomial(n+4,4) for n in (0..40)] # _G. C. Greubel_, Aug 28 2019

%o (GAP) List([0..40], n-> (n+1)*Binomial(n+4,4)); # _G. C. Greubel_, Aug 28 2019

%Y Partial sums of A002418.

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

%K nonn,easy

%O 0,2

%A Thi Ngoc Dinh (via _R. K. Guy_)