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Search: id:A099326
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| A099326 |
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Expansion of ((1-2x)sqrt(1+2x)+sqrt(1-2x))/(2(1-2x)^(5/2)). |
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+0
3
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| 1, 4, 11, 28, 67, 156, 354, 792, 1747, 3820, 8278, 17832, 38174, 81368, 172644, 365104, 769411, 1617228, 3389838, 7090440, 14797546, 30828424, 64106716, 133113168, 275967022, 571415416, 1181585564, 2440680592, 5035637212
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n)=sum{k=0..n, (k+1)binomial(n,(n-k)/2)binomial(k+3,3)(1+(-1)^(n-k))/(n+k+2)}. The g.f. is transformed to 1/(1-x)^4 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)^2, where c(x) is the g.f. of the Catalan numbers A000108.
0,1,4,11,28... is the image of the quarter-squares Floor((n+1)^2/4) (A002620(n+1)) under the Riordan array ((1+2x)/sqrt(1-4x^2), xc(x^2)). Hankel transform of A099326 has g.f. (1-x)/(1+x)^4. - Paul Barry (pbarry(AT)wit.ie), Oct 25 2007
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FORMULA
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a(n)=sum{k=0..n, (k+1)binomial(n, (n-k)/2)binomial(k+3, 3)(1+(-1)^(n-k))/(n+k+2)}.
a(n)=sum{k=0..n, C(n,k)*(Floor((abs(n-2k) + 1)^2/4)+Floor((abs(n-2k+1) + 1)^2/4)}; - Paul Barry (pbarry(AT)wit.ie), Oct 25 2007
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CROSSREFS
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Cf. A099325, A099327.
Sequence in context: A113478 A056601 A003230 this_sequence A127985 A005409 A020964
Adjacent sequences: A099323 A099324 A099325 this_sequence A099327 A099328 A099329
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Oct 12 2004
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