OFFSET
0,2
COMMENTS
a(n) is the number of compositions (ordered partitions) of n+1 into ten or fewer parts. - Geoffrey Critzer, Jan 24 2009
a(n) is the maximal number of regions in 9-space formed by n-1 8-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 1*0*1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859, A008860, A008861 count binary words of the form 0*1*0*, 1*0*1*0*, 0*1*0*1*0*, 1*0*1*0*1*0*, 0*1*0*1*0*1*0*, 1*0*1*0*1*0*1*0* and 0*1*0*1*0*1*0*1*0* respectively. - Manfred Scheucher, Jun 23 2023
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
a(n) = Sum_{k=1..5} binomial(n+1, 2*k-1), compare A008860.
From Geoffrey Critzer, Jan 24 2009: (Start)
G.f.: (1 - 8*x + 29*x^2 - 62*x^3 + 86*x^4 - 80*x^5 + 50*x^6 - 20*x^7 + 5*x^8)/(1-x)^10.
a(n) = (n^9 - 27*n^8 + 366*n^7 - 2646*n^6 + 12873*n^5 - 31563*n^4 + 79064*n^3 + 34236*n^2 + 270576*n + 362880)/9!. (End)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10); a(0)=1, a(1)=2, a(2)=4, a(3)=8, a(4)=16, a(5)=32, a(6)=64, a(7)=128, a(8)=256, a(9)=512. - Harvey P. Dale, Mar 18 2012
EXAMPLE
a(10)=1023 because there are (2^10)-1 compositions of 11 into ten or fewer parts. - Geoffrey Critzer, Jan 24 2009
MAPLE
seq(add(binomial(n, j), j=0..9), n=0..40); # G. C. Greubel, Sep 13 2019
MATHEMATICA
Table[Sum[Binomial[n, k], {k, 0, 9}], {n, 0, 40}] (* or *) LinearRecurrence[ {10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 2, 4, 8, 16, 32, 64, 128, 256, 512}, 40] (* Harvey P. Dale, Mar 18 2012 *)
PROG
(Haskell)
a008862 = sum . take 10 . a007318_row -- Reinhard Zumkeller, Nov 24 2012
(PARI) vector(40, n, sum(j=0, 9, binomial(n-1, j))) \\ G. C. Greubel, Sep 13 2019
(Magma) [(&+[Binomial(n, k): k in [0..9]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019
(Sage) [sum(binomial(n, k) for k in (0..9)) for n in (0..40)] # G. C. Greubel, Sep 13 2019
(GAP) List([0..40], n-> Sum([0..9], k-> Binomial(n, k)) ); # G. C. Greubel, Sep 13 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved