OFFSET
5,1
COMMENTS
Number of 16-subsequences of [ 1, n ] with just 5 contiguous pairs.
LINKS
T. D. Noe, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
FORMULA
G.f.: 42*x^5*(6 + 5*x)/(1-x)^12.
a(n) = C(n+1, 6)*C(n+5, 5). - Zerinvary Lajos, May 26 2005; corrected by R. J. Mathar, Feb 10 2016
E.g.f.: (42/10!)*x^5*(181440 + 206640*x + 82800*x^2 + 15300*x^3 + 1400*x^4 + 61*x^5 + x^6)*exp(x). - G. C. Greubel, Feb 09 2020
From Amiram Eldar, Feb 03 2022: (Start)
Sum_{n>=5} 1/a(n) = 10446907/211680 - 5*Pi^2.
Sum_{n>=5} (-1)^(n+1)/a(n) = 5*Pi^2/2 + 512*log(2)/21 - 8799517/211680. (End)
MAPLE
seq( 42*(n+1)*binomial(n+5, 10), n=5..30); # G. C. Greubel, Feb 09 2020
MATHEMATICA
Table[42(n+1)Binomial[n+5, 10], {n, 5, 30}] (* Harvey P. Dale, Feb 09 2019 *)
PROG
(PARI) vector(26, n, my(m=n+4); binomial(m+1, 6)*binomial(m+5, 5)) \\ G. C. Greubel, Feb 09 2020
(Magma) [Binomial(n+1, 6)*Binomial(n+5, 5): n in [5..30]]; // G. C. Greubel, Feb 09 2020
(Sage) [42*(n+1)*binomial(n+5, 10) for n in (5..30)] # G. C. Greubel, Feb 09 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Thi Ngoc Dinh (via R. K. Guy)
STATUS
approved