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A018213 Alkane (or paraffin) numbers l(12,n). 4

%I

%S 1,5,30,110,365,1001,2520,5720,12190,24310,46252,83980,147070,248710,

%T 408760,653752,1021735,1562275,2343770,3453450,5008003,7153575,

%U 10080720,14024400,19284460,26225628,35304920,47071640

%N Alkane (or paraffin) numbers l(12,n).

%C Equals (1/2) * ((A000582) + (A000332 interleaved with zeros)) = (1/2) * ((1, 10, 55, 220, 715...) + (1, 0, 5, 0, 15,...)); where A000582 = binomial(n,9) and A000332 = binomial(n,4).

%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

%D Winston C. Yang (paper in preparation).

%H Vincenzo Librandi, <a href="/A018213/b018213.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1).

%F l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.

%F G.f.: (5*x^4+10*x^2+1)/((x-1)^10*(x+1)^5). [_Colin Barker_, Aug 06 2012]

%F a(n) = (1/(2*9!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9) +(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n). [_Yosu Yurramendi_, Jun 23 2013]

%t CoefficientList[Series[(5 x^4 + 10 x^2 + 1)/((x - 1)^10 (x + 1)^5), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 16 2013 *)

%t LinearRecurrence[{5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1},{1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760},101] (* _Ray Chandler_, Sep 23 2015 *)

%o (MAGMA) [(1/(2*Factorial(9)))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)+(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n): n in [0..40]]; // _Vincenzo Librandi_, Oct 16 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)

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Last modified May 27 13:39 EDT 2018. Contains 304693 sequences. (Running on oeis4.)