A008863 a(n) = Sum_{k=0..10} binomial(n,k).

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4083, 8100, 15914, 30827, 58651, 109294, 199140, 354522, 616666, 1048576, 1744436, 2842226, 4540386, 7119516, 10970272, 16628809, 24821333, 36519556, 53009102, 75973189, 107594213, 150676186, 208791332

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 eleven or fewer parts. - Geoffrey Critzer, Jan 24 2009

a(n) is the maximal number of regions in 10-space formed by n-1 9-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 0*1*0*1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862 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*, 0*1*0*1*0*1*0*1*0* and 1*0*1*0*1*0*1*0*1*0* respectively. - Manfred Scheucher, Jun 23 2023

Examples

			a(11) = 2047 because there are 2^11=2048 compositions of 12 into any size parts but one of the compositions (1+1+...+1=12) has more than eleven parts. - _Geoffrey Critzer_, Jan 24 2009
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165, 55,-11,1).

Crossrefs

Cf. A008859, A008860, A008861, A008862, A006261, A000127, A007318, A219531.

Programs

GAP

List([0..40], n-> Sum([0..10], k-> Binomial(n,k)) ); # G. C. Greubel, Sep 13 2019

Haskell

a008863 = sum . take 11 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

Magma

[(&+[Binomial(n,k): k in [0..10]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019

Maple

A008863:=n->add(binomial(n,k), k=0..10): seq(A008863(n), n=0..40); # Wesley Ivan Hurt, Apr 28 2017

Mathematica

Table[Sum[Binomial[n, i], {i, 0, 10}], {n, 0, 40}] (* T. D. Noe, Mar 27 2012 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,2,4,8, 16,32,64,128,256,512,1024}, 40] (* Harvey P. Dale, Apr 25 2012 *)

PARI

a(n)=sum(k=0,10,binomial(n,k)) \\ Charles R Greathouse IV, Apr 07 2016

Python

A008863_list, m = [], [1, -8, 29, -62, 86, -80, 50, -20, 5, 0, 1]
for _ in range(10**2):
    A008863_list.append(m[-1])
    for i in range(10):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

Sage

[sum(binomial(n,k) for k in (0..10)) for n in (0..40)] # G. C. Greubel, Sep 13 2019

Formula

a(n) = Sum_{k=0..5} binomial(n+1, 2k), compare A008859.

From Geoffrey Critzer, Jan 24 2009: (Start)

G.f.: (1 - 9*x + 37*x^2 - 91*x^3 + 148*x^4 - 166*x^5 + 130*x^6 - 70*x^7 + 25*x^8 - 5*x^9 + x^10)/(1-x)^11.

a(n) = (n^10 - 35*n^9 + 600*n^8 - 5790*n^7 + 36813*n^6 - 140595*n^5 + 408050*n^4 - 382060*n^3 + 1368936*n^2 + 2342880*n + 3628800)/10!. (End)

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11); 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, a(10)=1024. - Harvey P. Dale, Apr 25 2012