A000039 -id:A000039 - cp's OEIS frontend

A000025 Coefficients of the 3rd-order mock theta function f(q).

1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.

			G.f. = 1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe and then corrected by Sean A. Irvine, Apr 25 2019)
George E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95.
Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
Leila A. Dragonette, Some asymptotic formulas for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015. [See f(q)]
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).
Ken Ono, The last words of a genius, Notices Amer. math. Soc., 57 (2010), 1410-1419.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
Eric Weisstein's World of Mathematics, Mock Theta Function.

Other '3rd-order' mock theta functions are at A013953, A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.

Cf. A340601, A340692.

a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
        (1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
seq(a(n), n=0..120);

CoefficientList[Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}], q] (* N. J. A. Sloane *)
sgn[P_ (* a partition *)] :=
Signature[
  PermutationList[
   Cycles[Flatten[
     SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@
       Accumulate[P]], Length[P] - 1]]]]
conjugate[P_List(* a partition *)] :=
Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];
  Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]
Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@
    IntegerPartitions[#]] & /@ Range[20]
(* George Beck, Oct 25 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / Product[ 1 + x^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]]; (* Michael Somos, Jun 30 2015 *)
rnk[prts_]:=Max[prts]-Length[prts]; mtf[n_]:=Module[{pn=IntegerPartitions[n]},Total[If[ EvenQ[ rnk[#]],1,-1]&/@pn]]; Join[{1},Array[mtf,60]] (* Harvey P. Dale, Sep 13 2024 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))}; /* Michael Somos, Sep 02 2007 */

my(N=60, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)^2/(1+x^k))) \\ Seiichi Manyama, May 23 2023

G.f.: 1 + Sum_{n>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).
G.f.: (1 + 4 * Sum_{n>=1} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>=1} (1 - q^i).
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - Vaclav Kotesovec, Jun 10 2019
G.f.: 1 - Sum_{n >= 1} (-1)^n*x^n/Product_{k = 1..n} 1 + x^k. See Fine, equation 26.22, p. 55. - Peter Bala, Feb 04 2021
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A340601(n) - A340692(n).
G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k)^2 / (1+x^k). (End)

Entry improved by comments from Dean Hickerson

A000199 Coefficient of q^(2n-1) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 3, 7, 6, 12, 13, 20, 21, 34, 36, 51, 58, 78, 89, 118, 131, 171, 197, 245, 279, 349, 398, 486, 557, 671, 767, 920, 1046, 1244, 1421, 1667, 1898, 2225, 2525, 2937, 3333, 3856, 4367, 5034, 5683, 6521, 7365, 8409, 9473, 10795, 12133, 13775, 15466

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..1001 from T. D. Noe)
L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
Eric Weisstein's World of Mathematics, Mock Theta Function.

A000025(2n-1)=a(n). Cf. A000039.

f[q_, s_] := Sum[q^(n^2)/Product[1+q^k, {k, n}]^2, {n, 0, s}]; Take[CoefficientList[Series[f[q, 100 ], {q, 0, 100}], q], {2, -1, 2}]

a(n)=if(n<1,0,polcoeff(1+sum(k=1,sqrtint(2*n-1),x^k^2/prod(i=1,k,1+x^i,1+O(x^(2*n-1)))^2),2*n-1))

a(n) ~ exp(Pi*sqrt(n/3)) / (2*sqrt(2*n)). - Vaclav Kotesovec, Jun 11 2019

More terms from Eric W. Weisstein

A294407 Expansion of 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^j)^2).

Original entry on oeis.org

1, -1, 3, -8, 20, -51, 132, -339, 868, -2228, 5720, -14676, 37659, -96644, 248004, -636413, 1633144, -4190920, 10754580, -27598012, 70821032, -181738372, 466370429, -1196782952, 3071141180, -7881051500, 20224069573, -51898276576, 133179482008, -341760374284, 877013123076, -2250559385788

Offset: 0

Ilya Gutkovskiy, Oct 30 2017

sign

Convolution inverse of the 3rd order mock theta function f(q) (A000025).

Robert Israel, Table of n, a(n) for n = 0..2441
Eric Weisstein's World of Mathematics, Mock Theta Function

Cf. A000025, A000039, A000199, A010815.

N:= 50: # to get a(0)..a(N)
g:= 1/(1+add(q^(i^2)/mul(1+q^j,j=1..i)^2, i=1..floor(sqrt(N)))):
S:= series(g, q, N+1):
seq(coeff(S,q,n),n=0..N); # Robert Israel, Nov 06 2017

nmax = 31; CoefficientList[Series[1/(1 + Sum[q^(i^2)/Product[(1 + q^j)^2, {j, 1, i}], {i, 1, nmax}]), {q, 0, nmax}], q]

G.f.: 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^j)^2).

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

A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m.

Original entry on oeis.org

6, 12, 18, 24, 28, 30, 36, 42, 48, 54, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 112, 114, 117, 120, 126, 132, 138, 140, 144, 150, 156, 162, 168, 174, 180, 182, 186, 192, 196, 198, 204, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282

Offset: 1

Jaroslav Krizek, Aug 30 2019

nonn

All integers m contain at least one divisor d (number 1) such that sigma(d) divides m.
See A309253 for the smallest numbers m with n divisors d such that sigma(d) divides m for n >= 1.
Supersequence of A097603 (multiples of perfect numbers).
From Bernard Schott, Sep 04 2019: (Start)
If m = 6 * k with k >= 1, then 2 divides m and sigma(2) = 3 also divides m; so, the positive multiples of 6 belong to this sequence.
This sequence is generated by the primitive terms. A primitive term m is necessarily of the form d * sigma(d) where 1 < d < m is a divisor of m. The first few primitives are: 6, 28, 117, 182, ...
Some subsequences of such primitives, not exhaustive list:
1) d is prime p and m = p * sigma(p) = p * (p+1) is oblong.
For p = 2, 13, 19, 37, ..., we get 6, 182, 380, 1406, ...
2) d = p^2 with p prime, and m = p^2 * (p^2 + p + 1).
For p = 2, 3, 5, 7, ..., we get m = 28, 117, 775, 2793, ...
3) d = 2^(q-1) and m = 2^(q-1) * (2^q -1), with q prime in A000043 and 2^q - 1 is a Mersenne prime in A000668, then m is a perfect number in A000039.
For q prime = 2, 3, 5, 7, 13, ..., we get m = 6, 28, 496, 8128, 33550336, ... (End)

			Divisors d of 12: 1, 2, 3, 4, 6, 12; corresponding sigma(d):1, 3, 4, 7, 12, 28; sigma(d) divides 12 for 4 divisors d > 1: 2, 3 and 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A173441, A309253, A323652, A326697, A326698.

Subsequences: A008588 \ {0}, A097603.

[m: m in [1..10^5] | #[d: d in Divisors(m) | IsIntegral(m / SumOfDivisors(d) ) and d gt 1] gt 0];

filter:= proc(n) local d;
  uses numtheory;
  ormap(t -> n mod sigma(t) = 0, divisors(n) minus {1})
end proc:
select(filter, [$2..1000]); # Robert Israel, Oct 07 2019

aQ[n_] := AnyTrue[Rest @ Divisors[n], Divisible[n, DivisorSigma[1, #]] &]; Select[Range[282], aQ] (* Amiram Eldar, Aug 31 2019 *)

isok(m) = fordiv(m, d, if ((d>1) && (!(m % sigma(d))), return(1))); \\ Michel Marcus, Sep 03 2019

A173441(a(n)) > 1; A326697(a(n)) > 1; A326697(a(n)) > 1.

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920

Offset: 0

Jonathan Vos Post, Aug 28 2005

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]

a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

cp's OEIS Frontend

A000025 Coefficients of the 3rd-order mock theta function f(q).

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A000199 Coefficient of q^(2n-1) in the series expansion of Ramanujan's mock theta function f(q).

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A294407 Expansion of 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^j)^2).

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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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A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m.

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Magma

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A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

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