OFFSET
0,1
COMMENTS
If Y is a 4-subset of an n-set X then, for n >= 11, a(n-11) is the number of 8-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
G.f.: (4-3*x)/(1-x)^9.
a(n) = 4*b(n) - 3*b(n-1) = (n+32)*binomial(n+7, 7)/8, with b(n) = binomial(n+8, 8) = A000581(n+8, 8).
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9), with a(0)=4, a(1)=33, a(2)=153, a(3)=525, a(4)=1485, a(5)=3663, a(6)=8151, a(7)=16731, a(8)=32175. - Harvey P. Dale, Jul 07 2015
From Amiram Eldar, Oct 21 2025: (Start)
Sum_{n>=0} 1/a(n) = 6881171814243790897/23732498287511763750.
Sum_{n>=0} (-1)^n/a(n) = 25940992*log(2)/202275 - 300618377711891653801/3390356898215966250. (End)
E.g.f.: (1/8!)*(161280 + 1169280*x + 1834560*x^2 + 1081920*x^3 + 294000*x^4 + 39984*x^5 + 2744*x^6 + 88*x^7 + x^8)*exp(x). - G. C. Greubel, Jan 04 2026
MATHEMATICA
CoefficientList[Series[(4-3x)/(1-x)^9, {x, 0, 30}], x] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {4, 33, 153, 525, 1485, 3663, 8151, 16731, 32175}, 30] (* Harvey P. Dale, Jul 07 2015 *)
PROG
(Maxima) A095671(n):=(n+32)*binomial(n+7, 7)/8$
makelist(A095671(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
(Magma)
A095671:= func< n | (n+32)*Binomial(n+7, 7)/8 >;
[A095671(n): n in [0..40]]; // G. C. Greubel, Jan 04 2026
(SageMath)
def A095671(n): return (n+32)*binomial(n+7, 7)//8
print([A095671(n) for n in range(41)]) # G. C. Greubel, Jan 04 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 11 2004
STATUS
approved