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A219615 a(n) = Sum_{k=0..12} binomial(n,k). 2
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16369, 32647, 64839, 127858, 249528, 480492, 910596, 1695222, 3096514, 5546382, 9740686, 16777216, 28354132, 47050564, 76717268, 123012781, 194129627, 301766029, 462411533, 699030226, 1043243132 (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 thirteen or fewer parts.

a(n) is the sum of the first thirteen terms in the n-th row of Pascal's triangle.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

FORMULA

a(n) = (n^12 - 54n^11 + 1397n^10 - 21450n^9 + 218823n^8 - 1508562n^7 + 7374191n^6 - 23551110n^5 + 58206676n^4 - 48306984n^3 + 173699712n^2 + 312888960n)/479001600. - Charles R Greathouse IV, Nov 27 2012

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(11)=2048, a(12)=4096, a(n)= 13*a(n-1)- 78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+ 1716*a(n-7)- 1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - Harvey P. Dale, Nov 29 2012

EXAMPLE

a(13)= 8191 because there are (2^13) -1 compositions of 14 into thirteen or fewer parts. When 1<= n <= 12, for n=5, a(5) = 2*a(4) = 2*16 = 32. For n=12, a(12) = 2*a(11)= 2*2048 = 4096. When n>12, for n=13, a(13) = 2*a(12) - binomial(12,12) = 2*4096 - 1 = 8191. For n = 15, a(15) = 2*a(14) - binomial(14,12) = 2*16369 - 91 = 32738 - 91 = 32647.

MATHEMATICA

Table[Sum[Binomial[n, k], {k, 0, 12}], {n, 0, 40}] (* T. D. Noe, Nov 27 2012 *)

LinearRecurrence[{13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1}, {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096}, 40] (* Harvey P. Dale, Nov 29 2012 *)

PROG

(PARI) a(n)=sum(k=1, 12, binomial(n, k)) \\ Charles R Greathouse IV, Nov 27 2012

CROSSREFS

Cf. A000127, A006261, A008859, A008860, A008861, A008862, A008863, A219531.

Sequence in context: A219531 A168083 A221180 * A168084 A133025 A216095

Adjacent sequences:  A219612 A219613 A219614 * A219616 A219617 A219618

KEYWORD

nonn,easy

AUTHOR

Mokhtar Mohamed, Nov 23 2012

EXTENSIONS

Sequence corrected and extended by T. D. Noe, Nov 26 2012

Definition corrected by Harvey P. Dale, Nov 29 2012

STATUS

approved

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Last modified June 24 12:30 EDT 2021. Contains 345416 sequences. (Running on oeis4.)