A215066 -id:A215066 - cp's OEIS frontend

A053250 Coefficients of the '3rd-order' mock theta function phi(q).

1, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 1, -1, -2, 1, 3, -1, -2, 1, 2, -2, -3, 1, 4, 0, -4, 2, 3, -2, -4, 1, 5, -2, -5, 3, 5, -3, -5, 2, 7, -2, -7, 3, 6, -4, -8, 3, 9, -2, -9, 5, 9, -5, -10, 3, 12, -4, -12, 5, 11, -6, -13, 6, 16, -6, -15, 7, 15, -8, -17, 7, 19, -6, -20, 9, 19, -10, -22, 8, 25, -9, -25, 12, 25, -12, -27, 11, 31

Offset: 0

Dean Hickerson, Dec 19 1999

			G.f. = 1 + x - x^3 + x^4 + x^5 - x^6 - x^7 + 2*x^9 - 2*x^11 + x^12 + x^13 - x^14 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.12), p. 58, Eq. (26.56).
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

T. D. Noe, Table of n, a(n) for n = 0..1000
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.
A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From _N. J. A. Sloane_, Feb 09 2013
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053251, A053252, A053253, A053254, A053255.

Cf. A207569, A215066.

f:=n->q^(n^2)/mul((1+q^(2*i)),i=1..n); add(f(n),n=0..10);

Series[Sum[q^n^2/Product[1+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x^2, x^2, k], {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = my(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)) + O(x^(n - (k-1)^2 + 1)), 1), n))}; /* Michael Somos, Jul 16 2007 */

Consider partitions of n into distinct odd parts. a(n) = number of them for which the largest part minus twice the number of parts is == 3 (mod 4) minus the number for which it is == 1 (mod 4).
a(n) = (-1)^n*(A027358(n)-A027357(n)). - Vladeta Jovovic, Mar 12 2006
G.f.: 1 + Sum_{k>0} x^k^2 / ((1 + x^2) (1 + x^4) ... (1 + x^(2*k))).
G.f. 1 + Sum_{n >= 0} x^(2*n+1)*Product_{k = 1..n} (x^(2*k-1) - 1) (Folsom et al.). Cf. A207569 and A215066. - Peter Bala, May 16 2017

A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))(1 - exp(-3t)) + (1 - exp(-t))(1 - exp(-3t))(1 - exp(-5t)) + ... as a series in t.

Original entry on oeis.org

1, 1, 5, 55, 1073, 32671, 1431665, 85363615, 6646603073, 654896692351, 79656194515025, 11722538113191775, 2052949879753739873, 421931472111868912831, 100568330857984368195185

Offset: 0

Peter Bala, Mar 24 2009

We appear to get the same sequence by expanding 1 - (1 - exp(t)) + (1 - exp(t))*(1 - exp(2*t)) - (1 - exp(t))*(1 - exp(2*t))*(1 - exp(3*t)) + ... as a series in t. Compare with A079144. For other sequences with generating functions of a similar type see A000364, A000464, A002105 and A002439.
From Peter Bala, Mar 13 2017: (Start)
It appears that the g.f. has two other forms: either F(exp(-t)) where F(q) = Sum_{n >= 0} q^(n+1)*Product_{k = 1..n} 1 - q^(2*k) = q + q^2 + q^3 - q^7 - q^8 - q^10 - q^11 - ... is a g.f. for A003475 or 1/2*G(exp(t)) where G(q) = 1 + Sum_{n >= 0} (-1)^n*q^(n+1)*Product_{k = 1..n} 1 - q^k = 1 + q - q^2 + 2*q^3 - 2*q^4 + q^5 + q^7 - 2*q^8 + ... is a g.f. for A003406. See Zagier, Example 1. (End)
From Peter Bala, Dec 18 2021: (Start)
Conjectures:
1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 16 begins [1, 1, 5, 7, 1, 15, 1, 15, 1, 15, 1, 15, ...] with an apparent pre-period of length 4 and a period of length 2.
2) Let i >= 0 and define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.
If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients. For example, the expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 21*x^3 + 291*x^4 + 6861*x^5 + 246171*x^6 + 12458901*x^7 + 843915891*x^8 + 73640674461*x^9 + 8041227405771*x^10 + ... appears to have integer coefficients. (End)

			G.f. A(x) = 1 + x + 5*x^2 + 55*x^3 + 1073*x^4 + 32671*x^5 + 1431665*x^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..170
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
D. Zagier, Quantum modular forms, Quanta of Maths: Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675

Cf. A000364, A000464, A002105, A002439, A079144, A158691, A209832, A215066, A003406, A003475.

max = 14; se = Series[1 + Sum[ Product[1 - E^(-(2*k - 1)*t), {k, 1, n}], {n, 1, max}], {t, 0, max}]; CoefficientList[se, t]*Range[0, max]! (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, 1-exp(-(2*k-1)*x+x*O(x^n)))), n)} \\ Paul D. Hanna, Aug 01 2012

{a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, exp(k*x+x*O(x^n))-1)), n)} \\ Paul D. Hanna, Aug 01 2012

Basic hypergeometric generating function: 1 + Sum_{n >= 0} Product_{k = 1..n} (1 - exp(2*k-1)*t) = 1 + t + 5*t^2/2! + 55*t^3/3! + ....
a(n) ~ 6*sqrt(2) * 12^n * (n!)^2 / Pi^(2*n+2). - Vaclav Kotesovec, May 04 2014
Conjectural g.f.: G(exp(t)) as a formal power series in t, where G(q) := Sum_{n >= 0} q^(2*n+1) * Product_{k = 1..2*n} (1 - q^k). - Peter Bala, May 16 2017
E.g.f.: Sum_{n>=0} exp(n*(n+1)/2*x) / Product_{k=0..n} (1 + exp(k*x)). - Paul D. Hanna, Oct 14 2020

A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

Original entry on oeis.org

1, 1, 3, 18, 151, 1640, 21825, 343763, 6253234, 128993019, 2975165831, 75866604098, 2119310099700, 64361149952242, 2111222815441491, 74391641880144734, 2802300974537717340, 112379709083552152423, 4780136025081921948194, 214954914688567198802759

Offset: 0

Paul D. Hanna, Feb 18 2012

Compare g.f. to Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

Compare g.f. to Sum_{n>=0} Product_{k=1..n} (1 - (1 - x)^(2*k-1)), which is the g.f. of A158691. - Peter Bala, Dec 04 2020

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 151*x^4 + 1640*x^5 + 21825*x^6 + ...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1) + ...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207557, A207570, A207571, A215066, A053250, A158691.

CoefficientList[Series[Sum[Product[(1+x)^(2*k-1)-1, {k, 1, n}], {n, 0, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, May 06 2014 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ sqrt(12) * 24^n * n^n / (exp(n+Pi^2/48) * Pi^(2*n+1)). - Vaclav Kotesovec, May 06 2014
G.f.: 1/2*( 1 + Sum_{n>=0} (1 + x)^(2*n+1) * Product_{k = 1..n} ((1 + x)^(2*k-1) - 1) ). Cf. A053250 and A215066. - Peter Bala, May 15 2017
Conjectural g.f.: Sum_{n>=0} (-1)^n*Product_{k = 1..n} 1 + ( -1/(1 + x) )^k. - Peter Bala, Dec 04 2020
From Peter Bala, Jan 29 2021: (Start)
Conjectural g.f.s: Sum_{n >= 0} (-1)^n*(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k*(1 + x)^k)^2. Also
(1/2)*( 1 + Sum_{n >= 0} 1/(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k/(1 + x)^k) ). (End)

A214687 E.g.f.: Sum_{n>=0} exp(2nx) * Product_{k=1..n} (exp((2k-1)x) - 1).

Original entry on oeis.org

1, 1, 11, 217, 7691, 430921, 35117531, 3927676537, 577640740331, 108115035641641, 25097054302205051, 7076531411753120857, 2382432541064412524171, 943997056642739165681161, 434864796716131476530668571, 230460477665217932140097413177

Offset: 0

Paul D. Hanna, Aug 01 2012

nonn

Compare the e.g.f. to the identity:

exp(-x) = Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (1 - exp((2*k-1)*x)).

			E.g.f.: A(x) = 1 + x + 11*x^2/2! + 217*x^3/3! + 7691*x^4/4! + 430921*x^5/5! +...
such that, by definition,
A(x) = 1 + exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)
+ exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...
Compare this series to the identity:
exp(-x) = 1 - exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)
- exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)  +-...
The related e.g.f. of A215066 equals the series:
G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(3*x)-1)
+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...
or, more explicitly,
G(x) = 1 + x + 7*x^2/2! + 127*x^3/3! + 4315*x^4/4! + 235831*x^5/5! +...
such that G(x) satisfies:
G(x) = (1 + exp(x)*A(x))/2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..175
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A207214, A215066.

{a(n)=n!*polcoeff(sum(m=0,n+1,exp(2*m*x+x*O(x^n))*prod(k=1,m,exp((2*k-1)*x+x*O(x^n))-1)),n)}
for(n=0,26,print1(a(n),", "))

E.g.f. A(x) satisfies: A(x) = exp(-x)*(2*G(x) - 1),
where G(x) = Sum_{n>=0} Product_{k=1..n} (exp((2*k-1)*x) - 1) = e.g.f. of A215066.
a(n) ~ 2*sqrt(6) * 24^n * (n!)^2 / (sqrt(n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 05 2014

A218261 E.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2k-1)x) - 1) / (2*k-1).

Original entry on oeis.org

1, 1, 3, 19, 191, 2731, 52063, 1264747, 37912143, 1368247627, 58312623743, 2889264152875, 164299982895535, 10607439707069323, 770371122097072863, 62438253016068932203, 5608567981763102915087, 554952834214350689736139, 60161153106358242206145343

Offset: 0

Paul D. Hanna, Oct 24 2012

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2731*x^5/5! +...
where
A(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(3*x)-1)/(1*3) + (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)/(1*3*5) + (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)/(1*3*5*7) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..165

Cf. A215066, A135752.

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,(exp((2*k-1)*X)-1)/(2*k-1)));n!*polcoeff(Egf,n)}
for(n=0,30,print1(a(n),", "))

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A053250 Coefficients of the '3rd-order' mock theta function phi(q).

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A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))*(1 - exp(-3*t)) + (1 - exp(-t))*(1 - exp(-3*t))*(1 - exp(-5*t)) + ... as a series in t.

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A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

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A214687 E.g.f.: Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (exp((2*k-1)*x) - 1).

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A218261 E.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2*k-1)*x) - 1) / (2*k-1).

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A158690 Expansion of the basic hypergeometric series 1 + (1 - exp(-t)) + (1 - exp(-t))(1 - exp(-3t)) + (1 - exp(-t))(1 - exp(-3t))(1 - exp(-5t)) + ... as a series in t.

A214687 E.g.f.: Sum_{n>=0} exp(2nx) * Product_{k=1..n} (exp((2k-1)x) - 1).

A218261 E.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2k-1)x) - 1) / (2*k-1).