Search: id:A003106
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%I A003106 M0261
%S A003106 1,0,1,1,1,1,2,2,3,3,4,4,6,6,8,9,11,12,15,16,20,22,26,29,35,38,45,50,58,
               64,75,
%T A003106 82,95,105,120,133,152,167,190,210,237,261,295,324,364,401,448,493,551,
%U A003106 604,673,739,820,899,997,1091,1207,1321,1457,1593,1756,1916,2108,2301
%N A003106 Number of partitions of n into parts 5k+2 or 5k+3.
%C A003106 Expansion of Rogers-Ramanujan function H(x) in powers of x.
%C A003106 Also number of partitions of n such that the number of parts is greater 
               by one than the smallest part. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Mar 04 2006
%C A003106 Example: a(10)=4 because we have [9, 1], [6, 2, 2], [5, 3, 2] and [4, 
               4, 2]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006
%C A003106 Also number of partitions of n such that if the largest part is k, then 
               there are exactly k-1 parts equal to k. Example: a(10)=4 because 
               we have [3, 3, 2, 2], [3, 3, 2, 1, 1], [3, 3, 1, 1, 1, 1] and [2, 
               1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 09 2006
%C A003106 Also number of partitions of n such that if the largest part is k, then 
               k occurs at least k+1 times. Example: a(10)=4 because we have [2, 
               2, 2, 2, 2], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1] and [1, 1, 
               1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 09 2006
%C A003106 Also number of partitions of n such that the smallest part is larger 
               than the number of parts. Example: a(10)=4 because we have [10], 
               [7, 3], [6, 4] and [5, 5]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 09 2006
%D A003106 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003106 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 238.
%D A003106 G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 
               44 (No. 4, 2007), 561-573.
%D A003106 G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University 
               Press, 1999; Exercise 6(f), p. 591.
%D A003106 G. E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan 
               identities. Amer. Math. Monthly 96 (1989), no. 5, 401-409.
%D A003106 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 108.
%D A003106 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), 
               no. 8, 697-712.
%H A003106 T. D. Noe, <a href="b003106.txt">Table of n, a(n) for n=0..1000</a>
%H A003106 P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/hep-th/0505097">
               Parafermionic derivation of Andrews-type multiple-sums</a>
%H A003106 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rogers-RamanujanIdentities.html">Link to a section of The World of 
               Mathematics.</a>
%F A003106 The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n*(n+1))/((1-t)*(1-t^2)*...*(1-t^n)) 
               = Product_{n >= 1} 1/((1-t^(5*n-2))*(1-t^(5*n-3))); this is the g.f. 
               for the sequence.
%F A003106 G.f.: (Product_{k>0} 1+x^(2k))*(Sum_{k>=0} x^(k^2+2k)/(Product_{i=1..k} 
               1-x^(4i))). - Michael Somos Oct 19 2006
%F A003106 Euler transform of period 5 sequence [ 0, 1, 1, 0, 0, ...]. - Michael 
               Somos Oct 15 2008
%e A003106 a(10)=4 because we have [8, 2], [7, 3], [3, 3, 2, 2] and [2, 2, 2, 2, 
               2].
%e A003106 q^11 + q^131 + q^191 + q^251 + q^311 + 2*q^371 + 2*q^431 + 3*q^491 + 
               ...
%p A003106 g:=1/product((1-x^(5*j-2))*(1-x^(5*j-3)),j=1..15): gser:=series(g,x=0,
               66): seq(coeff(gser,x,n),n=0..63); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 09 2006
%o A003106 (PARI) {a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, 
               (sqrtint(4*n + 1) - 1) \ 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n 
               - k^2 - k))), 1), n))} /* Michael Somos Oct 15 2008 */
%Y A003106 Cf. A003114.
%Y A003106 Sequence in context: A058747 A050365 A029026 this_sequence A026824 A025149 
               A026799
%Y A003106 Adjacent sequences: A003103 A003104 A003105 this_sequence A003107 A003108 
               A003109
%K A003106 nonn,nice,easy
%O A003106 0,7
%A A003106 N. J. A. Sloane (njas(AT)research.att.com), Herman P. Robinson

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