A207569 -id:A207569 - 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.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

A215066 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2k-1)x) - 1).

Original entry on oeis.org

1, 1, 7, 127, 4315, 235831, 18911467, 2091412807, 305035062955, 56729101908151, 13102338649018027, 3679320979659518887, 1234515698986458346795, 487763952468349266962071, 224150079034073231822617387, 118541831524545132821950527367

Offset: 0

Paul D. Hanna, Aug 01 2012

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 127*x^3/3! + 4315*x^4/4! + 235831*x^5/5! +...
where
A(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) + (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)*(exp(9*x)-1) +...

Michael De Vlieger, Table of n, a(n) for n = 0..234
A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From _N. J. A. Sloane_, Feb 09 2013
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A053250, A158690, A207569, A214687.

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&*[Exp((2*k-1)*x) -1: k in [1..j]]): j in [1..m+1]]) )); [1] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 07 2020

m:= 20; S:= series( add(mul(exp((2*k-1)*x)-1, k=1..j), j=0..m+1), x, m+1): seq(factorial(j)*coeff(S, x, j), j = 0..m); # G. C. Greubel, Feb 07 2020

Table[((-1)^n*2*Sum[Sum[n!/(a!*(2b)!*(n-a-2b)!)*(3/2)^a*(5/2)^(2b) * EulerE[2a+2b],{a,0,n}],{b,0,n/2}] + 2*(-1)^n*Sum[n!/((n-2b)!*(2b)!)*(3/2)^(n-2b)*(1/2)^(2b)*EulerE[2n-2b],{b,0,n/2}])/4,{n,0,20}] (* Vaclav Kotesovec, May 04 2014 after A. Folsom *)
With[{m=20}, CoefficientList[Series[Sum[Product[Exp[(2*k-1)*x] -1, {k, j}], {j, 0, m+2}], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 07 2020 *)

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

m=20;
def A215066_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sum(product(exp((2*k-1)*x)-1 for k in (1..j)) for j in (0..m)) ).list()
a=A215066_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Feb 07 2020

Folsom et al. give a closed form for a(n). - N. J. A. Sloane, Feb 09 2013
E.g.f.: 1 + (exp(x)-1)/(W(0)-exp(x)+1), where W(k) = (exp(x))^(2*k+1) - ((exp(x))^(2*k+3)-1)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 05 2014
a(n) ~ sqrt(6) * 24^n * (n!)^2 / (sqrt(n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 04 2014
E.g.f.: 1/2*( 1 + Sum_{n>=0} exp((2*n+1)*x)*Product_{k=1..n} (exp((2*k-1)*x) - 1) ). Cf. A053250 and A207569. - Peter Bala, May 15 2017
Conjectural g.f.: Sum_{n >= 0} (-1)^n*Product_{k = 1..n} (1 + (-1)^k*exp(- k*t)). Cf. A158690. - Peter Bala, Jan 28 2021

A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

Original entry on oeis.org

1, 1, 4, 34, 410, 6455, 125251, 2888305, 77157780, 2342972405, 79701049425, 3002132647515, 124039845584382, 5577660227565634, 271162541308698623, 14172237715785139175, 792418822364402364530, 47198077739119663907870, 2983413619934353599892285

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

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

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 410*x^4 + 6455*x^5 + 125251*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^4-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1)*((1+x)^10-1) +...

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

Cf. A179525, A207556, A207569, A207571.

Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Product[ (1+x)^(3k-2)-1,{k,n}],{n,nn}],{x,0,nn}],x]]]] (* Harvey P. Dale, Aug 20 2012 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(3*k-2)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(2/3) * 2^(2*n-1/3) * 3^(2*n+5/6) * n^(n-1/6) / (exp(n+Pi^2/72) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 06 2014

A207571 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-1) - 1).

Original entry on oeis.org

1, 2, 11, 105, 1390, 23520, 484247, 11742927, 327711230, 10343198878, 364237027076, 14156867852699, 601927703437645, 27790427952836499, 1384496764982434033, 74027620787319243688, 4228343290201028904807, 256946673653717460509502, 16551666142815138743519611

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

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

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 105*x^3 + 1390*x^4 + 23520*x^5 + 484247*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)^2-1) + ((1+x)^2-1)*((1+x)^5-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1)*((1+x)^11-1) +...

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

Cf. A179525, A207556, A207569, A207570.

CoefficientList[Series[Sum[Product[(1+x)^(3*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)^(3*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(1/3) * 2^(2*n+1/3) * 3^(2*n+7/6) * n^(n+1/6) / (exp(n+Pi^2/72) * Pi^(2*n+11/6)). - Vaclav Kotesovec, May 06 2014

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

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A215066 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2*k-1)*x) - 1).

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A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

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A207571 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-1) - 1).

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A215066 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} (exp((2k-1)x) - 1).