A219676 a(n) = Sum_{k=0..13} binomial(n, k).

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16383, 32752, 65399, 130238, 258096, 507624, 988116, 1898712, 3593934, 6690448, 12236830, 21977516, 38754732, 67108864, 114159428, 190876696, 313889477, 508019104, 809785133, 1272196666

Offset: 0

Author

Mokhtar Mohamed, Nov 24 2012

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions (ordered partitions) of n+1 into fourteen or fewer parts.

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

Examples

			a(14) = 16383 because there are 2^14 = 16384 compositions of 15 into any size parts but one of the compositions (1 + 1 + ... + 1 = 15) has more than fourteen parts.
When 1 <= n <= 13, a(7) = 2*a(6) = 2*64= 128, a(13) = 2*a(12) = 2*4096 = 8192.
When n > 13, a(14) = 2*a(13) - C(13, 13) = 2*8192 - 1 = 16383, a(15) = 2*a(14) - C(14, 13) = 2*16383 - 14 = 32766 - 14 = 32752.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

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

Programs

Maple

f:= n -> add(binomial(n,k),k=0..13):
map(f, [$0..100]); # Robert Israel, Mar 14 2018

Mathematica

Table[Sum[Binomial[n, k], {k, 0, 13}], {n, 0, 40}] (* T. D. Noe, Nov 26 2012 *)

Formula

a(n) = Sum_{k=1..7} binomial(n+1, 2k-1).

a(n) = 1 +(n^13 -65*n^12 +2015*n^11 -37609*n^10 +470613*n^9 -4081935*n^8 [phone]*n^7 -110205667*n^6 [phone]*n^5 -657328100*n^4 [phone]*n^3 [phone]*n^2 [phone]*n)/13!. - corrected by Mokhtar Mohamed, Dec 01 2012

G.f.: (1 - 12*x + 67*x^2 - 230*x^3 + 541*x^4 - 920*x^5 + 1163*x^6 - 1106*x^7 + 791*x^8 - 420*x^9 + 161*x^10 - 42*x^11 + 7*x^12)/(1-x)^14.

a(n) = 2*a(n-1), for 1 <= n <= 13, with a(0) = 1, a(n) = 2*a(n-1) - C(n-1, 13), for n > 13.

Extensions

Corrected and extended by T. D. Noe, Nov 26 2012