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A027800
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a(n) = (n+1)*binomial(n+4, 4).
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9
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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
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OFFSET
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0,2
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COMMENTS
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Number of 9-subsequences of [1, n] with just 4 contiguous pairs.
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
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REFERENCES
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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).
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LINKS
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FORMULA
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G.f.: (1+4*x)/(1-x)^6.
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
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EXAMPLE
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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
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MAPLE
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a:=n->(n+1)^2*(n+2)*(n+3)*(n+4)/24: seq(a(n), n=0..40); # Emeric Deutsch
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MATHEMATICA
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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 *)
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PROG
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(PARI) vector(40, n, n*binomial(n+3, 4)) \\ G. C. Greubel, Aug 28 2019
(Magma) [(n+1)*Binomial(n+4, 4): n in [0..40]]; // G. C. Greubel, Aug 28 2019
(Sage) [(n+1)*binomial(n+4, 4) for n in (0..40)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..40], n-> (n+1)*Binomial(n+4, 4)); # G. C. Greubel, Aug 28 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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