OFFSET
0,2
COMMENTS
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved