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

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 *)

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

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

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

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

CROSSREFS

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

Sequence in context: A292568 A354600 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 17:52 EST 2022. Contains 358535 sequences. (Running on oeis4.)