A096661 -id:A096661 - cp's OEIS frontend

A097042 G.f. = (1 + 4 * g.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).

1, 2, 0, 4, 2, 4, 4, 8, 8, 10, 12, 16, 20, 24, 28, 36, 42, 48, 60, 72, 84, 100, 116, 136, 160, 186, 216, 252, 292, 336, 388, 448, 512, 588, 672, 768, 878, 1000, 1136, 1292, 1464, 1656, 1876, 2120, 2388, 2696, 3032, 3408, 3832, 4298, 4816, 5396, 6036, 6744, 7532, 8404

Offset: 0

N. J. A. Sloane, Sep 15 2004

nonn

a(0) = 1; for n>0, a(n) = 2*A026832(n) (i.e., essentially Fine's numbers L(n) multiplied by 2).

The number of odd-even overpartitions of n: an odd-even overpartition of n is an overpartition of n with the smallest part odd and such that the difference between successive parts is odd if the smaller part is nonoverlined and even otherwise - see Yang 2017. - Peter Bala, Mar 29 2017

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Min-Joo Jang, Asymptotic behavior of odd-even partitions, arXiv:1703.01837v1 [math.NT], 2017.

Cf. A096661, A026832, A179049.

nmax = 60; Flatten[{1, Rest[CoefficientList[Series[2*Sum[x^(2*k - 1) QPochhammer[-x^(2*k), x], {k, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Mar 28 2017 *)

a(n) ~ 1/(3^(5/4)*n^(3/4))*exp(Pi*sqrt(n/3)) [Jang 2017]. - Peter Bala, Mar 29 2017

Conjectural g.f.: 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ). - Peter Bala, Feb 19 2021

Name corrected by Peter Bala, Feb 19 2021

A026832 Number of partitions of n into distinct parts, the least being odd.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372, 3766, 4202, 4682

Offset: 0

Clark Kimberling

Fine's numbers L(n).

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1], [2,2,2,1], [2,1,1,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 29 2006

			a(7)=4 because we have [7], [6,1], [4,3] and [4,2,1].

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026804, A026805, A026807, A026833, A092265, A096661, A097042.

Cf. A000009, A096765.

a026832 n = p 1 n where
   p _ 0 = 1
   p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012

g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, b(n$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]]  (* Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
Join[{0},Table[Length[Select[IntegerPartitions[n],OddQ[#[[-1]]]&&Max[Tally[#][[All,2]]] == 1&]],{n,60}]] (* Harvey P. Dale, May 14 2022 *)

G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.: Sum_{k>=1} x^(2k-1)*Product_{j>=2k} (1 + x^j). - Emeric Deutsch, Mar 29 2006
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

More terms from Emeric Deutsch, Mar 29 2006

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019

A064053 Auxiliary sequence gamma(n) used to compute coefficients in series expansion of the mock theta function f(q) via A(n) = Sum_{r=0..n} p(r)*gamma(n-r), with p(r) the partition function A000041.

Original entry on oeis.org

1, 0, -4, 4, -4, 4, -4, 8, -4, 0, -4, 8, -4, 0, -4, 4, -4, 0, 0, 8, -4, -4, -4, 8, 0, 0, 0, 4, -4, 0, -4, 8, -4, -4, 0, 8, 0, 0, -8, 4, -8, 0, 4, 8, -4, 0, -8, 8, 0, 0, -4, 4, -4, 0, -4, 12, -4, 0, 0, 8, -4, 0, -8, 0, -4, 4, 4, 8, -4, 0, -12, 8, 0, 0, 0, 4, -4, -4, -4, 8, -8, 0, 0, 8, 4, 4, -8, 0, -4, 0, 0, 4, -4, 0, -8, 12, 0, 0, 4, 0, -4, 0, -4

Offset: 0

Eric W. Weisstein, Aug 28 2001

sign

See Dragonette for the definition of f(q) and A(n). - N. J. A. Sloane, Sep 24 2022

			G.f. = 1 - 4*x^2 + 4*x^3 - 4*x^4 + 4*x^5 - 4*x^6 + 8*x^7 - 4*x^8 - 4*x^10 + 8*x^11 - 4*x^12 - ...

G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998, page 82, Example 5. MR1634067 (99c:11126). [The Gamma function used by Andrews is the classical Gamma function, which is different from the gamma(n) of this sequence. - N. J. A. Sloane, Sep 24 2022]

G. C. Greubel, Table of n, a(n) for n = 0..5000
L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. See page 496.
L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. Enlargement of a portion of page 496 in order to show correct spelling of gamma(n).
Eric Weisstein's World of Mathematics, Mock Theta Function.

Cf. A000025, A000039, A000041, A000199, A096661.

a[ n_]:= SeriesCoefficient[1 +4 *Sum[(-1)^k*x^(k*(3*k+1)/2)/(1+x^k), {k, Quotient[Sqrt[1 +24*n] - 1, 6]}], {x, 0, n}]; (* Michael Somos, Apr 08 2015 *)

{a(n) = if( n<1, n==0, 4 * polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^((3*k^2 + k)/2) / (1 + x^k), x * O(x^n)), n))}; /* Michael Somos, Mar 13 2006 */

G.f.: 1 + 4 * Sum_{k>0} (-1)^k * x^(k*(3*k + 1)/2) / (1 + x^k). - Michael Somos, Jun 19 2003
Convolution of this sequence and A000041 is A000025. - Michael Somos, Jun 19 2003
a(n) = 4 * A096661(n) unless n=0.

Entry revised by Michael Somos, Mar 13 2006

Deleted edit that tried to change gamma(n) to Gamma(n), and restored original definition. - N. J. A. Sloane, Sep 24 2022

cp's OEIS Frontend

A097042 G.f. = (1 + 4 * g.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).

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A026832 Number of partitions of n into distinct parts, the least being odd.

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A064053 Auxiliary sequence gamma(n) used to compute coefficients in series expansion of the mock theta function f(q) via A(n) = Sum_{r=0..n} p(r)*gamma(n-r), with p(r) the partition function A000041.

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