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A220051
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Sum_{k=0..14} binomial(n,k).
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1
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1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32767, 65519, 130918, 261156, 519252, 1026876, 2014992, 3913704, 7507638, 14198086, 26434916, 48412432, 87167164, 154276028, 268435456, 459312152, 773201629, 1281220733, 2091005866
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of compositions (ordered partitions) of n+1 into fifteen or fewer parts.
a(n) = sum(binomial(n+1,2k), for k = 0..7).
a(n) is the sum of the first fifteen terms in the n-th row of Pascal's triangle.
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LINKS
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FORMULA
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a(n) = 1 + (n^14 - 77*n^13 + 2821*n^12 - 6288*n^11 + 947947*n^10 - 10081071*n^9 + 77889383*n^8 - 435638203*n^7 + 1793239448*n^6 - 5043110072*n^5 + 1111159696*n^4 - 8346754416*n^3 + 30605906304*n^2 + 57424792320*n)/14!.
G.f.: (1 - 13x + 79x^2 - 297x^3 + 771x^4 - 1461x^5 + 2083x^6 - 2269x^7 + 1897x^8 - 1211x^9 + 581x^10 - 203x^11 + 49x^12 - 7x^13 + x^14)/(1-x)^15.
a(n) = 2*a(n-1), for 1 <= n <= 14, with a(0) = 1, a(n) = 2*a(n-1) - C(n-1,14), for n> 14.
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EXAMPLE
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a(15) = 32767 because there are 2^15 = 32768 compositions of 16 into any size parts but one of the compositions (1 + 1 + ... + 1 = 16) has more than fifteen parts.
When 1 <= n <= 14, for n=10, a(10) = 2*a(9) = 2*512 = 1024. For n=14, a(14) = 2*a(13) = 2*8192 = 16384.
When n > 14, for n = 15, a(15) = 2*a(14) -C(14,14) = 2*16384 -1 = 32767. For n=20, a(20) = 2*a(19) -C(19,14) = 2*519252 -11626 = 1038504 -11626 = 1026876.
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MATHEMATICA
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Table[Sum[Binomial[n, k], {k, 0, 14}], {n, 0, 33}] (* Indranil Ghosh, Feb 22 2017 *)
NestList[{#1 + 1, 2 #2 - Boole[#1 >= 14] Binomial[#1, 14]} & @@ # &, {0, 1}, 33][[All, -1]] (* Michael De Vlieger, Feb 22 2017 *)
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PROG
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(PARI) a(n)=sum(k=0, 14, binomial(n, k)) \\ Indranil Ghosh, Feb 23 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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