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0, 1, 6, 21, 60, 155, 378, 889, 2040, 4599, 10230, 22517, 49140, 106483, 229362, 491505, 1048560, 2228207, 4718574, 9961453, 20971500, 44040171, 92274666, 192937961, 402653160, 838860775, 1744830438, 3623878629, 7516192740
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OFFSET
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0,3
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COMMENTS
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Sum_n a(n)*k^n/n! = 2*exp(2k)-exp(k) = 12.059830... = Sum_n A000225(n+1)*k^n/n! = Sum_n A053545(n)*k^n/n!
a(n)/2^n is the expected value of the cardinality of the generalized union of n randomly selected (with replacement) subsets of [n] where the probability of selection is equal for all subsets. [From Geoffrey Critzer, May 18 2009]
Form a triangle in which interior members T(i,j)=T(i-1,j-1)+ T(i-1,j). The exterior members are given by 1,2,3,...2*n-1:T(1,1)=n, T(2,1)=n-1, T(3,1)=n-2...T(n,1)=1 and T(2,2)=n+1, T(3,3)=n+2,...T(n,n)=2*n-1. The sum of all members will reproduce this sequence. For example with n=4 the exterior members are 1 to 7: row(1)=4; row(2)=3,5; row(3)=2,8,6; row(4)=1,10,14,7. The sum of all these members is 60, the fourth term in the sequence. - J. M. Bergot, Oct 16 2012
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REFERENCES
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A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,250
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FORMULA
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a(n) = 2a(n-1)+2^n = A000225(n)*A001477(n) = A036289(n)-A001477(n). G.f. x(1-2x^2)/((1-x)(1-2x))^2.
a(n)=sum(sum(C(n,j), j=1..n),k=1..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 10 2007
Row sums of triangles A132751. - Gary W. Adamson, Aug 28 2007
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EXAMPLE
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a(4)=4*(2^4-1)=4*15=60
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MAPLE
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a:=n->sum(n*binomial(n, k), k=1..n): seq(a(n), n=0..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007
a:=n->sum(sum(binomial(n, j), j=1..n), k=1..n): seq(a(n), n=0..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 10 2007
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MATHEMATICA
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Table[n*2^n-n, {n, 0, 3*4!}] [From Vladimir Joseph Stephan Orlovsky, Apr 25 2010]
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PROG
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(Sage) [gaussian_binomial(n, 1, 2)*n for n in xrange(0, 29)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009]
(PARI) { for (n=0, 250, write("b066524.txt", n, " ", n*(2^n - 1)) ) } [From Harry J. Smith, Feb 21 2010]
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CROSSREFS
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Cf. A132751.
Sequence in context: A053809 A047520 A143115 * A113070 A009147 A012593
Adjacent sequences: A066521 A066522 A066523 * A066525 A066526 A066527
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, Jan 08 2002
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STATUS
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approved
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