The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A053261 Coefficients of the '5th-order' mock theta function psi_1(q). 17
 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Number of partitions of n such that each part occurs at most twice and if k occurs as a part then all smaller positive integers occur. Strictly unimodal compositions with rising range 1, 2, 3, ..., m where m is the largest part and distinct parts in the falling range (this follows trivially from the comment above). [Joerg Arndt, Mar 26 2014] REFERENCES 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 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz) George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134. George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255. William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence. George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304. FORMULA G.f.: psi_1(q) = Sum_{n>=0} q^(n*(n+1)/2) * Product_{k=1..n} (1 + q^k). a(n) ~ sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0, add(b(n-i*j, i+1), j=1..min(2, n/i)))) end: a:= n-> b(n, 1): seq(a(n), n=0..100); # Alois P. Heinz, Mar 26 2014 MATHEMATICA Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}] (* Second program: *) b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, Sum[b[n - i*j, i + 1], {j, 1, Min[2, n/i]}]]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *) nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *) PROG (PARI) N = 66; x = 'x + O('x^N); gf = sum(n=0, N, x^(n*(n+1)/2) * prod(k=1, n, 1+x^k) ); v = Vec(gf) /* Joerg Arndt, Apr 21 2013 */ CROSSREFS Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267. Sequence in context: A165640 A082892 A025839 * A123584 A291983 A112689 Adjacent sequences: A053258 A053259 A053260 * A053262 A053263 A053264 KEYWORD nonn,easy AUTHOR Dean Hickerson, Dec 19 1999 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 21:40 EST 2022. Contains 358570 sequences. (Running on oeis4.)