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A008863 Sum C(n,k), k=0..10. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the number of compositions (ordered partitions) of n+1 into eleven or fewer 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).

FORMULA

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

G.f.: (1-9x+37x^2-91x^3+148x^4-166x^5+130x^6-70x^7+25x^8-5x^9+x^10) / (1-x)^11.  a(n) = (n^10 -35n^9 +600n^8 -5790n^7 +36813n^6 -140595n^5 +408050n^4 -382060n^3 +1368936n^2 +2342880n +3628800)/10! [From Geoffrey Critzer, Jan 24 2009]

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, 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). - Harvey P. Dale, Apr 25 2012

EXAMPLE

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

MATHEMATICA

Table[Sum[Binomial[n, i], {i, 0, 10}], {n, 0, 100}] (* 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}, 100] (* Harvey P. Dale, Apr 25 2012 *)

PROG

(Haskell)

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

(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

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

CROSSREFS

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

Sequence in context: A243088 A113010 A056767 * A145117 A172320 A234592

Adjacent sequences:  A008860 A008861 A008862 * A008864 A008865 A008866

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified December 3 20:40 EST 2016. Contains 278745 sequences.