Search: id:A099326
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%I A099326
%S A099326 1,4,11,28,67,156,354,792,1747,3820,8278,17832,38174,81368,172644,
%T A099326 365104,769411,1617228,3389838,7090440,14797546,30828424,64106716,
%U A099326 133113168,275967022,571415416,1181585564,2440680592,5035637212
%N A099326 Expansion of ((1-2x)sqrt(1+2x)+sqrt(1-2x))/(2(1-2x)^(5/2)).
%C A099326 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.
%C A099326 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
%F A099326 a(n)=sum{k=0..n, (k+1)binomial(n, (n-k)/2)binomial(k+3, 3)(1+(-1)^(n-k))/
               (n+k+2)}.
%F A099326 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
%Y A099326 Cf. A099325, A099327.
%Y A099326 Sequence in context: A113478 A056601 A003230 this_sequence A127985 A005409 
               A020964
%Y A099326 Adjacent sequences: A099323 A099324 A099325 this_sequence A099327 A099328 
               A099329
%K A099326 easy,nonn
%O A099326 0,2
%A A099326 Paul Barry (pbarry(AT)wit.ie), Oct 12 2004

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