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

1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760, 653752, 1021735, 1562275, 2343770, 3453450, 5008003, 7153575, 10080720, 14024400, 19284460, 26225628, 35304920, 47071640

Offset

0,2

Comments

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).

References

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

Winston C. Yang (paper in preparation).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Classic Sequences

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

Index entries for linear recurrences with constant coefficients, signature (5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1).

Formula

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.

G.f.: (5*x^4+10*x^2+1)/((x-1)^10*(x+1)^5). [Colin Barker, Aug 06 2012]

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]

Mathematica

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

Prog

(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

Crossrefs

Sequence in context: A174002 A030506 A062990 * A359975 A047661 A000649

Adjacent sequences: A018210 A018211 A018212 * A018214 A018215 A018216

Keyword

nonn,easy

Author

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved