|
|
A006261
|
|
a(n) = Sum_{k=0..5} binomial(n,k).
(Formerly M1126)
|
|
38
|
|
|
1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, 206368, 242825, 284274, 331212, 384168, 443704, 510416, 584935, 667928, 760099, 862190
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. - Geoffrey Critzer, Jan 19 2009
Also the interpolating polynomial for the divisors of 32: {a(k): 0 <= k < 6} = {1,2,4,8,16,32}. - Reinhard Zumkeller, Jun 17 2009
a(n) is the maximal number of regions in 5-space formed by n-1 4-dimensional hypercubes. - Carl Schildkraut, May 26 2015
a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*1*0*. A000124, A000125, A000127 count binary words of the form 0*1*0*, 1*0*1*0*, and 0*1*0*1*0*, respectively. - Manfred Scheucher, Jun 22 2023
|
|
REFERENCES
|
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = binomial(n+1, 5) + binomial(n+1, 3) + binomial(n+1, 1). - Len Smiley, Oct 20 2001
G.f.: (1 - 4*x + 7*x^2 - 6*x^3 + 3*x^4)/(1-x)^6. - Geoffrey Critzer, Jan 19 2009
E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120)*exp(x).
|
|
EXAMPLE
|
a(7) = 120 because the first six terms in the 7th row of Pascal's triangle 1 + 7 + 21 + 35 + 35 + 21 = 120. - Geoffrey Critzer, Jan 19 2009
|
|
MAPLE
|
|
|
MATHEMATICA
|
CoefficientList[
Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,
52}], x]*Table[n!, {n, 0, 52}]
|
|
PROG
|
(Sage) [binomial(n, 1)+binomial(n, 3)+binomial(n, 5) for n in range(1, 38)] # Zerinvary Lajos, May 17 2009
(Magma) [(n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011
(Haskell)
(Python)
A006261_list, m = [], [1, -3, 4, -2, 1, 1]
for _ in range(10**2):
for i in range(5):
|
|
CROSSREFS
|
Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A007318, A008859, A008860, A008861, A008862, A008863, A219531.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|