A008861 a(n) = Sum_{k=0..8} binomial(n,k).

1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1013, 1981, 3797, 7099, 12911, 22819, 39203, 65536, 106762, 169766, 263950, 401930, 600370, 880970, 1271626, 1807781, 2533987, 3505699, 4791323, 6474541, 8656937, 11460949, 15033173, 19548046

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions (ordered partitions) of n+1 into nine or fewer parts. - Geoffrey Critzer, Jan 24 2009

a(n) is the maximal number of regions in 8-space formed by n-1 7-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 0*1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859, A008860 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*, and 1*0*1*0*1*0*1*0* respectively. - Manfred Scheucher, Jun 22 2023

Examples

			a(9)=511 because all but one (namely 1+1+1+...+1=10) of the 2^9 compositions of 10 are in nine or fewer parts. - _Geoffrey Critzer_, Jan 24 2009
		

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 (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A008859, A008860, A008862, A008863, A006261, A000127.

Cf. A007318, A219531.

Programs

GAP

List([0..40], n-> Sum([0..8], k-> Binomial(n,k)) ); # G. C. Greubel, Sep 13 2019

Haskell

a008861 = sum . take 9 . a007318_row -- Reinhard Zumkeller, Nov 24 2012

Magma

[(&+[Binomial(n,k): k in [0..8]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019

Maple

seq(sum(binomial(n,j), j=0..8), n=0..40); # G. C. Greubel, Sep 13 2019

Mathematica

Sum[Binomial[Range[41]-1, j-1], {j,9}] (* G. C. Greubel, Sep 13 2019 *)

PARI

vector(40, n, sum(j=0,8, binomial(n-1,j))) \\ G. C. Greubel, Sep 13 2019

Sage

[sum(binomial(n,k) for k in (0..8)) for n in (0..40)] # G. C. Greubel, Sep 13 2019

Formula

a(n) = Sum_{k=0..4} binomial(n+1, 2*k), compare A008859.

From Geoffrey Critzer, Jan 24 2009: (Start)

G.f.: (1 - 7*x + 22*x^2 - 40*x^3 + 46*x^4 - 34*x^5 + 16*x^6 - 4*x^7 + x^8)/(1-x)^9.

a(n) = (n^8 - 20*n^7 + 210*n^6 - 1064*n^5 + 3969*n^4 - 4340*n^3 + 15980*n^2 + 25584*n + 40320)/8!. (End)