Search: id:A027187
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%I A027187
%S A027187 1,0,1,1,3,3,6,7,12,14,22,27,40,49,69,86,118,146,195,242,317,392,505,
%T A027187 623,793,973,1224,1498,1867,2274,2811,3411,4186,5059,6168,7427,9005,
%U A027187 10801,13026,15572,18692,22267,26613,31602,37619,44533,52815,62338
%N A027187 Number of partitions of n into an odd number of parts, the least being 
               1; also a(n+1) = number of partitions of n into an even number of 
               parts; also partitions of n+1 in which the greatest part is even.
%D A027187 N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
%D A027187 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 8, (7.323) and p. 39, Example 7.
%H A027187 T. D. Noe, <a href="b027187.txt">Table of n, a(n) for n = 1..1000</a>
%F A027187 a(n+1)=(A000041(n)+(-1)^n*A000700(n))/2.
%F A027187 a(n+1)=p(n)-p(n-1)+p(n-4)-p(n-9)+... where p(n) is the number of unrestricted 
               partitions of n, A000041. [Fine] - David Callan (callan(AT)stat.wisc.edu), 
               Mar 14 2004
%F A027187 G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3 q^4 + 3 q^5 + 
               6 q^6 + ...
%F A027187 = Sum_{n >= 0} q^(2n)/(q; q)_{2n}
%F A027187 = ((Prod_{k >= 1} 1/(1-q^k) + (Prod_{k >= 1} 1/(1+q^k))/2 (R. William 
               Gosper (rwg(AT)osots.com), Jun 25 2005)
%F A027187 Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2 q^3 + 2 q^4 
               + 4 q^5 + 5 q^6 + ...
%F A027187 Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Prod_{k >= 1} 1/(1-q^k) 
               - (Prod_{k >= 1} 1/(1+q^k))/2.
%F A027187 Also we have the following identity involving 2 X 2 matrices:
%F A027187 Prod_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]
%F A027187 = [ A(q) B(q) / B(q) A(q) ] (R. William Gosper (rwg(AT)osots.com), Jun 
               25 2005)
%F A027187 a(2*n)=A046682(2*n), a(2*n+1)=A000701(2*n+1); a(n)=A000041(n)-A027193(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
%F A027187 Expansion of (1+phi(q))/(2f(-q)) where phi(),f() are Ramanujan theta 
               functions.
%F A027187 G.f.: (Sum_{k>=0} (-1)^n x^n^2)/(Product_{k>0}(1-x^k)). - Michael Somos 
               Aug 19 2006
%e A027187 a(1) = 1 from the partition 1 = 1; a(2) = 0; a(3) = 1 from 3 = 1+1+1.
%o A027187 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^oo); polcoeff( sum(k=0,sqrtint(n), 
               (-x)^k^2, A)/eta(x+A), n))} /* Michael Somos Aug 19 2006 */
%Y A027187 Cf. A027193, A000701, A046682.
%Y A027187 Cf. A026838.
%Y A027187 Sequence in context: A083751 A034401 A088571 this_sequence A056508 A050065 
               A078477
%Y A027187 Adjacent sequences: A027184 A027185 A027186 this_sequence A027188 A027189 
               A027190
%K A027187 nonn
%O A027187 1,5
%A A027187 Clark Kimberling (ck6(AT)evansville.edu)
%E A027187 This is an example of a sequence where there are two good choices for 
               the offset. Offset 1 is consistent with the first part of the definition. 
               - N. J. A. Sloane (njas(AT)research.att.com), Aug 27 2006

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