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A008863 a(n) = Sum_{k=0..10} binomial(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
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
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
LINKS
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(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
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
Sequence in context: A292568 A354600 A056767 * A145117 A172320 A234592
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)