Search: a168342 -id:a168342
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A011782
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Coefficients of expansion of (1-x)/(1-2*x) in powers of x.
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+10
872
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1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592
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OFFSET
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0,3
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COMMENTS
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Apart from initial term, same as A000079 (powers of 2).
Also the number of compositions (ordered partitions) of n. - Toby Bartels, Aug 27 2003
Number of ways of putting n unlabeled items into (any number of) labeled boxes where every box contains at least one item. Also "unimodal permutations of n items", i.e., those which rise then fall. (E.g., for three items: ABC, ACB, BCA and CBA are unimodal.) - Henry Bottomley, Jan 17 2001
Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani Albert Tshifhumulo, Apr 20 2001. More generally (see Simion and Schmidt), the number of permutations in S_n avoiding (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 321 patterns; (x) the 312 and 321 patterns.
a(n+2) is the number of distinct Boolean functions of n variables under action of symmetric group.
Also the number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003
Image of the central binomial coefficients A000984 under the Riordan array ((1-x), x*(1-x)). - Paul Barry, Mar 18 2005
Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051. - Philippe Deléham, Jul 04 2005
Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan, May 29 2006
Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51, ...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16, ...) = A(x)/A(x^2). - Gary W. Adamson, Feb 18 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. - Johannes W. Meijer, Aug 15 2010
Array T(m,n) = 2*T(m,n-1) + T(m-1,n):
1, 1, 2, 4, 8, 16, ... = a(n)
1, 3, 8, 20, 48, 112, ... = A001792,
1, 5, 18, 56, 160, 432, ... = A001793,
1, 7, 32, 120, 400, 1232, ... = A001794,
1, 9, 50, 220, 840, 2912, ... = A006974, followed with A006975, A006976, gives nonzero coefficients of Chebyshev polynomials of first kind A039991 =
1,
1, 0,
2, 0, -1,
4, 0, -3, 0,
8, 0, -8, 0, 1.
T(m,n) third vertical: 2*n^2, n positive (A001105).
Fourth vertical appears in Janet table even rows, last vertical (A168342 array, A138509, rank 3, 13, = A166911)). (End)
0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16, = a(n),
0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16.
Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. - David W. Wilson, Jan 01 2012
These could be called the composition numbers (see the second comment) since the equivalent sequence for partitions is A000041, the partition numbers. - Omar E. Pol, Aug 28 2013
Number of self conjugate integer partitions with exactly n parts for n>=1. - David Christopher, Aug 18 2014
The sequence is the INVERT transform of (1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). - Gary W. Adamson, Jul 16 2015
Also, number of threshold graphs on n nodes [Hougardy]. - Falk Hüffner, Dec 03 2015
Number of ternary words of length n in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
a(n) is the number of words of length n over an alphabet of two letters, of which one letter appears an even number of times (the empty word of length 0 is included). See the analogous odd number case in A131577, and the Balakrishnan reference in A006516 (the 4-letter odd case), pp. 68-69, problems 2.66, 2.67 and 2.68. - Wolfdieter Lang, Jul 17 2017
Number of D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are D-equivalent iff the positions of pattern D are identical in these paths. - Sergey Kirgizov, Apr 08 2018
Number of color patterns (set partitions) for an oriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if we permute the colors. For a(4)=8, the 4 achiral patterns are AAAA, AABB, ABAB, and ABBA; the 4 chiral patterns are the 2 pairs AAAB-ABBB and AABA-ABAA. - Robert A. Russell, Oct 30 2018
The determinant of the symmetric n X n matrix M defined by M(i,j) = (-1)^max(i,j) for 1 <= i,j <= n is equal to a(n) * (-1)^(n*(n+1)/2). - Bernard Schott, Dec 29 2018
For n>=1, a(n) is the number of permutations of length n whose cyclic representations can be written in such a way that when the cycle parentheses are removed what remains is 1 through n in natural order. For example, a(4)=8 since there are exactly 8 permutations of this form, namely, (1 2 3 4), (1)(2 3 4), (1 2)(3 4), (1 2 3)(4), (1)(2)(3 4), (1)(2 3)(4), (1 2)(3)(4), and (1)(2)(3)(4). Our result follows readily by conditioning on k, the number of parentheses pairs of the form ")(" in the cyclic representation. Since there are C(n-1,k) ways to insert these in the cyclic representation and since k runs from 0 to n-1, we obtain a(n) = Sum_{k=0..n-1} C(n-1,k) = 2^(n-1). - Dennis P. Walsh, May 23 2020
Maximum number of preimages that a permutation of length n + 1 can have under the consecutive-231-avoiding stack-sorting map. - Colin Defant, Aug 28 2020
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REFERENCES
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Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7
Xavier Merlin, Methodix Algèbre, Ellipses, 1995, p. 153.
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LINKS
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FORMULA
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a(0) = 1, a(n) = 2^(n-1).
G.f.: (1 - x) / (1 - 2*x) = 1 / (1 - x / (1 - x)). - Michael Somos, Apr 18 2012
E.g.f.: cosh(z)*exp(z) = (exp(2*z) + 1)/2.
a(0) = 1 and for n>0, a(n) = sum of all previous terms.
a(n) = Sum_{k=0..n} binomial(n, 2*k). - Paul Barry, Feb 25 2003
a(n) = Sum_{k=0..n} binomial(n,k)*(1+(-1)^k)/2. - Paul Barry, May 27 2003
a(n) = floor((1+2^n)/2). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
G.f.: Sum_{i>=0} x^i/(1-x)^i. - Jon Perry, Jul 10 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k+1, n-k)*binomial(2*k, k). - Paul Barry, Mar 18 2005
E.g.f.: (exp(2*x)+1)/2 = (G(0) + 1)/2; G(k) = 1 + 2*x/(2*k+1 - x*(2*k+1)/(x + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011
A051049(n) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
A008619(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 18 2012
G.f.: U(0), where U(k) = 1 + x*(k+3) - x*(k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
E.g.f.: E(0), where E(k) = 1 + x/( 2*k+1 - x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 25 2013
G.f.: 1 + x/(1 + x)*( 1 + 3*x/(1 + 3*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 7*x/(1 + 7*x)*( 1 + ... )))). - Peter Bala, May 27 2017
a(n) = Sum_{k=0..2} stirling2(n, k).
a(n) = A053120(n, n), n >= 0, (main diagonal of triangle of Chebyshev's T polynomials). - Wolfdieter Lang, Nov 26 2019
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EXAMPLE
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G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 16*x^5 + 32*x^6 + 64*x^7 + 128*x^8 + ...
( -1 1 -1)
det ( 1 1 1) = 4
( -1 -1 -1)
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MAPLE
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with(PolynomialTools): A011782:=seq(coeftayl((1-x)/(1-2*x), x = 0, k), k=0..10^2); # Muniru A Asiru, Sep 26 2017
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MATHEMATICA
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f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* Robert G. Wilson v, Jul 07 2006 *)
CoefficientList[ Series[(1-x)/(1-2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 07 2006 *)
Table[Sum[StirlingS2[n, k], {k, 0, 2}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 2^(n-1))};
(PARI) Vec((1-x)/(1-2*x) + O(x^30)) \\ Altug Alkan, Oct 31 2015
(Haskell)
a011782 n = a011782_list !! n
a011782_list = 1 : scanl1 (+) a011782_list
(Sage) [sum(stirling_number2(n, j) for j in (0..2)) for n in (0..35)] # G. C. Greubel, Jun 02 2020
(Python)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Lee D. Killough (killough(AT)wagner.convex.com)
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EXTENSIONS
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STATUS
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approved
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A160914
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Extended s-block elements for Janet table.
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+10
4
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1, 2, 3, 4, 11, 12, 19, 20, 37, 38, 55, 56, 87, 88, 119, 120, 169, 170, 219, 220, 291, 292, 363, 364, 461, 462, 559, 560, 687, 688, 815, 816, 977, 978, 1139, 1140, 1339, 1340, 1539, 1540, 1781, 1782, 2023, 2024, 2311, 2312, 2599, 2600, 2937, 2938, 3275, 3276
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OFFSET
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1,2
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COMMENTS
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See A168342. Must be included in A167268. From right to left, first vertical is A168380 from 1 to 8. Second vertical is A168380-1. In (1) page 12, introducing elements 93 to 120, Janet says that there is a probable 8th row. For row 8, he proposes, like for row 7, 32 elements (89 to 120). Page 16 he presents 4 blocks: first has 2*8 elements, second: 6*6, third: 10*4, fourth: 14*2. Today, blocks are s,p,d,f for Mendeleyev-Moseley-Seaborg 118 elements periodic table. See (2), (3), A173592 and A138509. In 1927, only 88 on the first 92 elements were known; 41 (1937 discovered), 61 (1947), 85 (1940) and 87 (1939) were missing. Since 2010 (117 discovered) the first 118 elements are known. Janet predicted only 120 elements.
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REFERENCES
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Charles JANET, La structure du Noyau de l'atome,considérée dans la Classification périodique, des éléments chimiques, 1927 (Novembre) N. 2 Beauvais, 67 pages, 3 leaflets.
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LINKS
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EXAMPLE
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The following is an s-block, 2*8=16 elements, i.e., a(n) written vertically, after p-block, 6*6, (A138469).
1 2
3 4
5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20
31 32 33 34 35 36 37 38
49 50 51 52 53 54 55 56
81 82 83 84 85 86 87 88
113 114 115 116 117 118 119 120
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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