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 A053251 Coefficients of the '3rd-order' mock theta function psi(q) 27
 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, 42, 46, 51, 57, 62, 68, 76, 83, 91, 101, 109, 120, 132, 143, 156, 171, 186, 202, 221, 239, 259, 283, 306, 331, 360, 388, 420, 455, 490, 529, 572, 616, 663, 716, 769, 827 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. Number of ways to express n as a partial sum 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+3+1+1. - Jon Perry, Jan 01 2004 Also number of partitions of n such that the 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, Mar 08 2006 Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - Seiichi Manyama, Mar 17 2018 For Emeric Deutsch's comment above, (1) this appears to be an alternately equal case of A122130, (2) the ordered version (compositions) is A239327, (3) allowing any length gives A351006, (4) the even-length version is A351007. - Gus Wiseman, Feb 25 2022 REFERENCES N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13). Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355. Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe) Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500. George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80. FORMULA G.f.: psi(q) = Sum_{n>=1} q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ). G.f.: Sum_{k>=1} q^k*Product_{j=1..k-1} (1+q^(2*j)) (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006 a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 09 2019 EXAMPLE q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ... From Seiichi Manyama, Mar 17 2018: (Start) n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) --+--------------------------+------------------------- 1 | (1)                      | (1) 2 | (2)                      | (2) 3 | (3)                      | (3) 4 | (4)                      | (4)   | (1, 3)                   | (1, 3/2) 5 | (5)                      | (5)   | (1, 4)                   | (1, 2) 6 | (6)                      | (6)   | (1, 5)                   | (1, 5/2) 7 | (7)                      | (7)   | (1, 6)                   | (1, 3)   | (2, 5)                   | (2, 5/2) 8 | (8)                      | (8)   | (1, 7)                   | (1, 7/2)   | (2, 6)                   | (2, 3) 9 | (9)                      | (9)   | (1, 8)                   | (1, 4)   | (2, 7)                   | (2, 7/2)   | (1, 3, 5)                | (1, 3/2, 5/3) (End) MAPLE f:=n->q^(n^2)/mul((1-q^(2*i+1)), i=0..n-1); add(f(i), i=1..6); # second Maple program: b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1,       b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2)     end: a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))): seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018 MATHEMATICA Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}] (* Second program: *) b[n_, i_] := b[n, i] = Function[s, If[n > s, 0, If[n == s, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]]]]][i*(i + 1)/2]; a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 17 2018, after Alois P. Heinz *) PROG (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 (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 */ CROSSREFS Other '3rd-order' mock theta functions are at A000025, A053250, A053252, A053253, A053254, A053255. Cf. A003475. Cf. A035363, A035457, A122129, A122130, A122134, A122135, A351003, A351005. Sequence in context: A029148 A067842 A164066 * A090184 A174575 A029057 Adjacent sequences:  A053248 A053249 A053250 * A053252 A053253 A053254 KEYWORD nonn,easy,changed AUTHOR Dean Hickerson, Dec 19 1999 EXTENSIONS More terms from Emeric Deutsch, Mar 08 2006 STATUS approved

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Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)