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A099325
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Expansion of (sqrt(1+2x)+sqrt(1-2x))/(2(1-2x)^(3/2)).
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7
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1, 3, 7, 16, 35, 76, 162, 344, 723, 1516, 3158, 6568, 13598, 28120, 57956, 119344, 245123, 503116, 1030542, 2109704, 4311786, 8808328, 17969372, 36644176, 74640430, 151985016, 309170332, 628741264, 1277540828, 2595198256
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The g.f. is transformed to 1/(1-x)^3 under the Chebyshev transformation A(x)->1/(1+x^2)A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x), where c(x) is the g.f. of the Catalan numbers A000108.
Image of 2n+1 under the Riordan array (1/sqrt(1-4x^2),xc(x^2)). Hankel transform is (n+1)*(-1)^n. - Paul Barry (pbarry(AT)wit.ie), Oct 06 2007
a(n) is the minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 1 to n+1, and each number in a higher row is the the sum of the two numbers directly below it. - Nathaniel Johnston, Apr 20 2011
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 0..1000
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FORMULA
| a(n) = sum{k=0..n, (k+1)*binomial(n, (n-k)/2)*binomial(k+2, 2)*(1+(-1)^(n-k))/(n+k+2)}.
a(n) = 2^n + sum{k=0..floor((n-1)/2), (2*n-4*k-1)*binomial(n, k)}. - Nathaniel Johnston, Apr 20 2011
a(n) = M^n*V topmost term. M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals, [1,0,0,0,...] as the main diagonal; and the rest zeros. V = the vector [1,2,3,...]. - Gary W. Adamson, Jan 30 2012
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MAPLE
| a:=proc(n)return 2^n+add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end:
seq(a(n), n=0..30); # Nathaniel Johnston, Apr 20 2011
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CROSSREFS
| Cf. A066411, A099326, A099327.
Sequence in context: A133124 A104004 A101509 * A026778 A023523 A065979
Adjacent sequences: A099322 A099323 A099324 * A099326 A099327 A099328
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Oct 12 2004
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