Search: id:A053251
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%I A053251
%S A053251 0,1,1,1,2,2,2,3,3,4,5,5,6,7,8,9,11,12,13,16,17,19,22,24,27,31,34,37,
%T A053251 42,46,51,57,62,68,76,83,91,101,109,120,132,143,156,171,186,202,221,
%U A053251 239,259,283,306,331,360,388,420,455,490,529,572,616,663,716,769,827
%N A053251 Coefficients of the '3rd order' mock theta function psi(q)
%C A053251 Number of different partial sums of 1+[1,3]+[1,5]+[1,7]+[1,9]+... E.g. 
               a(6)=2 because we have 6=1+1+1+1+1+1=1+1+3+1 - Jon Perry (perry(AT)globalnet.co.uk), 
               Jan 01 2004
%C A053251 Also number of partitions of n such that largest part occurs exactly 
               once and all the other parts occur exactly twice. Example: a(9)=4 
               because we have [9],[7,1,1],[5,2,2] and [3,2,2,1,1]. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Mar 08 2006
%D A053251 Leila A. Dragonette, Some asymptotic formulae for the mock theta functions 
               of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500
%D A053251 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 55, Eq. (26.13).
%D A053251 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
%D A053251 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, 
               Narosa Publishing House, New Delhi, 1988, p. 31
%D A053251 George N. Watson, The final problem: an account of the mock theta functions, 
               J. London Math. Soc., 11 (1936) 55-80
%F A053251 G.f.: psi(q) = sum for n >= 1 of q^n^2/((1-q)(1-q^3)...(1-q^(2n-1)))
%F A053251 a(n) = number of partitions of n into odd parts such that if a number 
               occurs as a part then so do all smaller positive odd numbers
%F A053251 G.f.=sum(q^k*product(1+q^(2j), j=1..k-1), k=1..infinity), (see the Fine 
               reference, p. 58, Eq. (26,53)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 08 2006
%e A053251 q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
%p A053251 f:=n->q^(n^2)/mul((1-q^(2*i+1)),i=0..n-1); add(f(i),i=1..6);
%t A053251 Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 
               100}]
%o A053251 (PARI) { n=20; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; 
               for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]+2*i-1)); 
               c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); 
               c } (Jon Perry)
%o A053251 (PARI) {a(n) = local(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, 
               sqrtint(n), t *= x^(2*k-1) / (1 - x^(2*k-1)) + O(x^(n-(k-1)^2+1))), 
               n))} /* Michael Somos Sep 04 2007 */
%Y A053251 Other '3rd order' mock theta functions are at A000025, A053250, A053252, 
               A053253, A053254, A053255.
%Y A053251 Cf. A003475.
%Y A053251 Sequence in context: A029148 A067842 A164066 this_sequence A090184 A029057 
               A087897
%Y A053251 Adjacent sequences: A053248 A053249 A053250 this_sequence A053252 A053253 
               A053254
%K A053251 nonn,easy
%O A053251 0,5
%A A053251 Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 19 1999
%E A053251 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 08 2006

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