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
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