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A008862 a(n) = Sum_{k=0..9} C(n,k). 14
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2036, 4017, 7814, 14913, 27824, 50643, 89846, 155382, 262144, 431910, 695860, 1097790, 1698160, 2579130, 3850756, 5658537, 8192524, 11698223, 16489546, 22964087, 31621024, 43081973, 58115146, 77663192, 102875128 (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 ten 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 (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(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(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). - 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

Cf. A008859, A008860, A008861, A008863, A006261, A000127.

Cf. A007318, A219531.

Sequence in context: A115213 A009714 A051535 * A145116 A172319 A234591

Adjacent sequences: A008859 A008860 A008861 * A008863 A008864 A008865

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified December 3 17:09 EST 2022. Contains 358535 sequences. (Running on oeis4.)