A106336 -id:A106336 - cp's OEIS frontend

A106335 Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.

3, 2, 2, 6, 2, 7, 6, 3, 2, 6, 9, 2, 1, 9, 1, 1, 3, 3, 0, 9, 6, 9, 8, 7, 1, 3, 8, 6, 7, 3, 9, 8, 3, 0, 2, 3, 3, 2, 2, 9, 0, 4, 2, 4, 3, 7, 4, 6, 7, 1, 7, 4, 5, 2, 1, 6, 0, 5, 6, 2, 0, 9, 1, 2, 4, 5, 5, 4, 8, 6, 2, 6, 7, 4, 1, 1, 1, 5, 0, 6, 4, 9, 7, 4, 7, 1, 2, 3, 7, 3, 9, 9, 1, 2, 2, 1, 4, 7, 8, 5, 3, 7, 1, 9, 0

Offset: 0

Paul D. Hanna, Apr 29 2005

The g.f. of A106336 equals (1/x)*Series_Reversion( x*eta(x)/eta(x^2)^2 ).

This constant is very close to 2^(3/2) / (3*sqrt(e*Pi)) = 0.3226276326921911330637735905807475397715626276499133673167401123748... - Vaclav Kotesovec, Aug 02 2017

			x/F(x)=0.322627632692191133096987138673983023322904243746717452160562...
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
so F(x) = 1.9873697211846841452692897833444126... (A106334)
at x = 0.6411803884299545796456448886283011... (A106333).

Cf. A106333, A106334, A106336.

digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); x0/f[x0] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)

A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2)); A106335=A106333/A106334

Constant equals the ratio x/F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

A366174 Decimal expansion of a constant related to the asymptotics of A106336.

Original entry on oeis.org

4, 9, 8, 3, 3, 4, 7, 9, 7, 9, 3, 3, 6, 0, 3, 4, 2, 2, 6, 0, 6, 3, 5, 9, 2, 6, 4, 0, 2, 8, 5, 0, 0, 1, 6, 4, 4, 3, 0, 6, 9, 4, 2, 8, 2, 3, 3, 0, 5, 1, 2, 9, 0, 2, 0, 1, 9, 9, 6, 8, 5, 3, 4, 9, 8, 3, 4, 0, 8, 7, 7, 4, 8, 5, 5, 2, 2, 7, 8, 4, 0, 2, 8, 9, 1, 1, 4, 7, 7, 3, 5, 2, 9, 2, 4, 3, 0, 3, 5, 0, 6, 2, 6, 0, 3, 1

Offset: 0

Vaclav Kotesovec, Oct 03 2023

			0.4983347979336034226063592640285001644306942823305129020199685349834...

Cf. A106334, A106335, A106336.

val = Sqrt[(s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2 * QPochhammer[-1, r*s]^2)/ (2*Pi*(-2*r^2*s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2* Derivative[0, 1][QPochhammer][-1, r*s]^2 + r*s*(-1 + r^2*s^2)*Log[r^2*s^2]*QPochhammer[-1, r*s]* ((Log[r^2*s^2] + 4*Log[1 - r^2*s^2] + 4*QPolyGamma[0, 1, r^2*s^2])* Derivative[0, 1][QPochhammer][-1, r*s] + r*s*Log[r^2*s^2] * Derivative[0, 2][QPochhammer][-1, r*s]) + 2*r^4*s^3*(-1 + r^2*s^2) * Log[r^2*s^2]^2*QPochhammer[-1, r*s]^3 * Derivative[0, 2][QPochhammer][r^2*s^2, r^2*s^2] + QPochhammer[-1, r*s]^2*(16*r^2*s^2*ArcTanh[1 - 2*r^2*s^2] + (-1 + r^2*s^2)*Log[r^2*s^2]^2 - 8*Log[1 - r^2*s^2] - 4*(-1 + r^2*s^2)* Log[1 - r^2*s^2]^2 - 8*(-1 + r^2*s^2)*(-1 + Log[1 - r^2*s^2])* QPolyGamma[0, 1, r^2*s^2] + (4 - 4*r^2*s^2)* QPolyGamma[0, 1, r^2*s^2]^2 + 4*(-1 + r^2*s^2)*(QPolyGamma[1, 1, r^2*s^2] - 2*r^2*s^2*Log[r^2*s^2]* Derivative[0, 0, 1][QPolyGamma][0, 1, r^2*s^2]))))] /. FindRoot[{2*s == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2], (-2*Log[1 - r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2])/ Log[r^2*s^2] + (r* s*(Derivative[0, 1][QPochhammer][-1, r*s] + r*QPochhammer[-1, r*s]^2* Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2]))/ QPochhammer[-1, r*s] == 1}, {r, 1/3}, {s, 2}, WorkingPrecision -> 600]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* r = A106335, s = A106334 *)

Equals limit_{n->oo} A106336(n) * n^(3/2) * A106335^n.

A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600

Offset: 0

Paul D. Hanna, Jun 18 2005

nonn

a(n) = Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch, May 15 2008

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
The g.f. A = A(x) also satisfies the identities:
(1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
(2) A(x) = 1 + x*A/(1-x*A) + x^2*A^2/((1-x*A)*(1-x^2*A^2)) + x^3*A^3/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3)) + ...
(3) A(x) = 1 + x*A/(1-x*A)^2 + x^4*A^4/((1-x*A)*(1-x^2*A^2))^2 + x^9*A^9/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3))^2 + ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(2*(1-x^2*A(x)^2)) + x^3*A(x)^3/(3*(1-x^3*A(x)^3)) + x^4*A(x)^4/(4*(1-x^4*A(x)^4)) + x^5*A(x)^5/(5*(1-x^5*A(x)^5)) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 + 12405*x^7/7 + 62337*x^8/8 + ... + A008485(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.5 and p.32
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications,arXiv:0805.1398v1 [math.CO], see p.5
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A109084, A184362, A008485, A000041; related: A106336.

A109085list[n_] := Module[{m = 1, A = 1 + x}, For[i = 1, i <= n, i++, A = 1/Product[(1 - x^k*(A + x*O[x]^n)^k), {k, 1, n}]]; CoefficientList[A, x][[1 ;; n]]]; A109085list[24] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)
InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *)

{a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,n,x^m*A^m/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,sqrtint(n+1),(x*A)^(m^2)/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)^2))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(x^m*A^m/m)/(1-x^m*A^m+x*O(x^n)) )));polcoeff(A,n)} \\ Paul D. Hanna, Jun 01 2011

{A008485(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A008485(m)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2012

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x*eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
(3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
(4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
(5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
(6) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^n/(1 - x^n*A(x)^n) ). - Paul D. Hanna, Jun 01 2011
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. - Paul D. Hanna, Feb 06 2012
a(n) = ([x^n] 1/((x; x)inf)^(n+1))/(n+1), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016

A106507 G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, -1, 1, -2, 3, -4, 5, -7, 10, -13, 16, -21, 28, -35, 43, -55, 70, -86, 105, -130, 161, -196, 236, -287, 350, -420, 501, -602, 722, -858, 1016, -1206, 1431, -1687, 1981, -2331, 2741, -3206, 3740, -4368, 5096, -5922, 6868, -7967, 9233, -10670, 12306, -14193

Offset: 0

Michael Somos, May 04 2005

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present entry gives 1/psi(q).

For various G.f. versions see the reciprocals of the ones given in A010054. - Wolfdieter Lang, Jul 05 2016

			G.f. = 1 - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 5*x^6 - 7*x^7 + 10*x^8 + ...
G.f. of B(q) =  A(q^8) / q = 1/q - q^7 + q^15 - 2*q^23 + 3*q^31 - 4*q^39 + 5*q^47 - 7*q^55 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 3rd equation, p. 41, 12th equation.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
C. Adiga, N. Anitha, T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A006950, A010054, A106336, A233758, A233759.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^((-1)^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, May 28 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) / EllipticTheta[ 2, 0, x^(1/2)] , {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] / QPochhammer[ x^4], {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)
(QPochhammer[x, x^2, 1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A)^2, n))};

Expansion of 1 / psi(x) in powers of x where psi() is a Ramanujan theta function, which is Jacobi's theta_2(0, sqrt(x))/(2*x^(1/8)) function. See, e.g., the Eric Weisstein link.
Expansion of q^(1/8) * eta(q) / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -1, 1, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4 * (w^4 + 4*v^4) - v^6*w^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2*u6^3 + u2^2*u3^3 - u3^3*u6^2.
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^3*u6^2 + 3*u1^3*u2^2 - u2^3*u3*u6.
G.f.: Sum_{k>=0} a(k) * x^(8*k - 1) = 1 / (Sum_{k in Z} x^((4k + 1)^2)).
G.f.: 1 / (1 + x + x^3 + x^6 + ...) = 1 - x * (1 - x) / (1 - x^2)^2 + x^4 * (1 - x) * (1 - x^2) / ((1 - x^2)^2 * (1 - x^4)^2) + ... [Ramanujan]  - Michael Somos, Jul 21 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A015128. - Michael Somos, Nov 01 2008
a(n) = (-1)^n * A006950(n). Convolution inverse of A010054.
Series reversion of A106336. - Michael Somos, May 10 2012
a(2*n) = A233758(n). a(2*n + 1) = - A233759(n). - Michael Somos, Nov 05 2015
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(2*k)). - Michael Somos, Nov 08 2015
G.f.: (x; x^2){1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017

Definition changed by N. J. A. Sloane, Aug 14 2007

A106337 Number of ways of writing n as the sum of n triangular numbers.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 82, 253, 757, 2173, 6341, 18888, 56266, 167324, 499773, 1499059, 4503557, 13546893, 40824379, 123233868, 372472353, 1127080252, 3414310032, 10353722919, 31425764410, 95463814056, 290222666436, 882954212908, 2688037654049, 8188468874808

Offset: 0

Paul D. Hanna, Apr 29 2005

nonn

Number of compositions of n into n triangular numbers with 0's allowed. a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - Alois P. Heinz, Jul 31 2017

The radius of convergence is equal to A106335. - Vaclav Kotesovec, Nov 15 2017

			G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.

Main diagonal of A286180.

b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017

QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017 *)

{a(n)=local(X); if(n<1,1,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n,n))}

Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).

a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - Alois P. Heinz, Aug 01 2017

a(0) changed to 1 by Alois P. Heinz, Jul 31 2017

A106334 Decimal expansion of the function F(x) evaluated at the constant x that satisfies: F(x) - xF'(x) = 0, where F(x) = Sum_{n>=0} x^(n(n+1)/2).

Original entry on oeis.org

1, 9, 8, 7, 3, 6, 9, 7, 2, 1, 1, 8, 4, 6, 8, 4, 1, 4, 5, 2, 6, 9, 2, 8, 9, 7, 8, 3, 3, 4, 4, 4, 1, 2, 6, 1, 8, 3, 4, 2, 7, 1, 7, 7, 2, 9, 8, 5, 5, 4, 5, 7, 4, 7, 0, 3, 5, 6, 2, 2, 3, 1, 0, 3, 8, 2, 6, 9, 5, 8, 9, 3, 8, 8, 6, 6, 2, 5, 5, 4, 7, 7, 6, 2, 0, 9, 7, 6, 2, 9, 9, 6, 3, 3, 6, 5, 7, 2, 7, 4, 6, 8, 1, 3, 5

Offset: 1

Paul D. Hanna, Apr 29 2005

Constant A106333 divided by this constant equals constant A106335, the radius of convergence of the g.f. of A106336.

			F(x)=1.9873697211846841452692897833444126183427177298554574703562231
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
at x = 0.6411803884299545796456448886283011... (A106333).

Cf. A106333, A106335, A106332, A106336.

digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); f[x0] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)

A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2))

A106333 Decimal expansion of the constant x that satisfies: F(x) - xF'(x) = 0, where F(x) = Sum_{n>=0} x^(n(n+1)/2).

Original entry on oeis.org

6, 4, 1, 1, 8, 0, 3, 8, 8, 4, 2, 9, 9, 5, 4, 5, 7, 9, 6, 4, 5, 6, 4, 4, 8, 8, 8, 6, 2, 8, 3, 0, 1, 1, 0, 6, 5, 5, 3, 4, 1, 9, 6, 1, 8, 9, 1, 0, 0, 7, 1, 1, 9, 0, 8, 7, 7, 5, 6, 0, 3, 0, 5, 0, 5, 1, 3, 1, 7, 2, 7, 8, 4, 5, 7, 5, 9, 2, 4, 7, 3, 3, 2, 3, 7, 8, 4, 6, 3, 5, 1, 2, 0, 8, 8, 3, 7, 9, 3, 2, 2, 4, 8, 9, 6

Offset: 0

Paul D. Hanna, Apr 29 2005

Not equal to exp(-4/9), which agrees with the first 16 decimal places. Related to Jacobi theta constant theta_2 and Dedekind's eta(x^2)^2/eta(x): Sum_{n>=0} x^(n*(n+1)/2) = 1.9873697... (A106334). This constant divided by constant A106334 equals constant A106335, the radius of convergence of the g.f. of A106336.

			0 = 1 - 2*x^3 - 5*x^6 - 9*x^10 - 14*x^15 - 20*x^21 - 27*x^28 - ...
x=0.641180388429954579645644888628301106553419618910071190877560305051317278

Cf. A106334, A106335, A106332, A106336.

digits = 105; g[x_?NumericQ] := NSum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 100]; x /. FindRoot[g[x], {x, 1/2}, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 12 2013 *)

solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2)))

Sum_{n>=0} (1 - n*(n+1)/2)*x^(n*(n+1)/2) = 0.

A194042 G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n(n+1)/2) )^8 where q=xA(x).

Original entry on oeis.org

1, 8, 92, 1248, 18590, 294032, 4848456, 82433472, 1434755717, 25438412696, 457838000316, 8342826818080, 153615385821902, 2853694780131056, 53419650732453288, 1006670308475537216, 19081647909610147674, 363577167520213046504, 6959630216092660689968

Offset: 0

Paul D. Hanna, Aug 12 2011

nonn

			G.f.: A(x) = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 +...
where
(0) A(x)^(1/8) = 1 + x*A(x) + x^3*A(x)^3 + x^6*A(x)^6 + x^10*A(x)^10 + x^15*A(x)^15 + x^21*A(x)^21 +...
(1) A(x)^(1/2) = 1/(1-x*A(x)) + 3*x*A(x)/(1-x^3*A(x)^3) + 5*x^2*A(x)^2/(1-x^5*A(x)^5) + 7*x^3*A(x)^3/(1-x^7*A(x)^7) +...
(2) A(x) = 1/(1-x^2*A(x)^2) + 8*x*A(x)/(1-x^4*A(x)^4) + 27*x^2*A(x)^2/(1-x^6*A(x)^6) + 64*x^3*A(x)^3/(1-x^8*A(x)^8) +...
(3) A(x) = (1+x*A(x))^8*(1-x^2*A(x)^2)^8 * (1+x^2*A(x)^2)^8*(1-x^4*A(x)^4)^8 * (1+x^3*A(x)^3)^8*(1-x^6*A(x)^6)^8 * (1+x^4*A(x)^4)^8*(1-x^8*A(x)^8)^8 *...
(4) log(A(x)) = 8*x*A(x)/(1+x*A(x)) + 8*(x^2*A(x)^2/(1+x^2*A(x)^2))/2  + 8*(x^3*A(x)^3/(1+x^3*A(x)^3))/3 + 8*(x^4*A(x)^4/(1+x^4*A(x)^4))/4 +...
Related expansions begin:
_ A(x)^(1/8) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 + 5007234*x^7 + 85341048*x^8 +...+  A194043(n)*x^n +...
_ A(x)^(1/2) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 + 1649446*x^6 + 27574712*x^7 + 473750970*x^8 +...+  A194044(n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..600

Cf. A106336, A194043, A194044.

(* Calculation of constant d: *) 1/r /. FindRoot[{2*s^(1/8) == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2, r^2*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s] / QPochhammer[-1, r*s] + r^2*s^(15/8) * QPochhammer[-1, r*s] * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] - 2*Log[1 - r^2*s^2]/Log[r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2]/Log[r^2*s^2] == 1/8}, {r, 1/20}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 17 2024 *)

{a(n)=local(A=1+x, T=sum(m=0, sqrtint(2*n+1), x^(m*(m+1)/2))+x*O(x^n)); A=(serreverse(x/T^8)/x); polcoeff(A, n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrt(2*n+1),(x*A+x*O(x^n))^(m*(m+1)/2))^8);polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+1)))^2);polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*(x*A)^m/(1-(x*A+x*O(x^n))^(2*m+2))));polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+(x*A)^m)*(1-(x*A)^(2*m)+x*O(x^n)))^8); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, 8*(x*A)^m/(1+(x*A)^m+x*O(x^n))/m))); polcoeff(A, n)}

Let q = x*A(x), then g.f. A(x) satisfies:
(1) A(x) = ( Sum_{n>=0} (2*n+1) * q^n/(1 - q^(2*n+1)) )^2,
(2) A(x) = Sum_{n>=0} (n+1)^3 * q^n/(1 - q^(2*n+2)),
(3) A(x) = Product_{n>=1} (1 + q^n)^8*(1 - q^(2*n))^8,
(4) A(x) = exp( Sum_{n>=1} 8*(q^n/(1 + q^n))/n ),
(5) A(x/F(x)^8) = F(x)^8 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
due to q-series identities.
a(n) ~ c * d^n / n^(3/2), where d = 20.757466132085824914671628173626682246438530051407107800521045715415164484946... and c = 1.12762897595401376103508613947431975160374771449653856610031074284717... - Vaclav Kotesovec, Oct 07 2020

A384829 G.f. satisfies A(x) = Sum_{n>=0} x^(n(n+1)/2) A(x)^(n*(n+1)).

Original entry on oeis.org

1, 1, 2, 6, 22, 87, 359, 1535, 6758, 30431, 139442, 648001, 3046730, 14467286, 69281190, 334211603, 1622568398, 7921905397, 38871120255, 191586353683, 948083155952, 4708743978840, 23463673225988, 117271827518778, 587744334759630, 2953138645722287, 14872864243128300, 75066312240321173

Offset: 0

Paul D. Hanna, Jun 13 2025

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 87*x^5 + 359*x^6 + 1535*x^7 + 6758*x^8 + 30431*x^9 + 139442*x^10 + 648001*x^11 + 3046730*x^12 + ...
where A(x) = 1 + x*A(x)^2 + x^3*A(x)^6 + x^6*A(x)^12 + x^10*A(x)^20 + x^15*A(x)^30 + ... + x^(n*(n+1)/2) * A(x)^(n*(n+1)) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A106336, A109085.

(* Calculation of constants {d,c}: *) {1/r, 2*(s/Sqrt[Pi*(-5 + 8*r^(7/8)*s^(3/4) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[r]*s])])} /. FindRoot[{2*r^(1/8)*s^(5/4) == EllipticTheta[2, 0, Sqrt[r]*s], 5*s^(1/4) == 2*r^(3/8) * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[r]*s]}, {r, 1/4}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 13 2025 *)

{a(n) = my(A = sqrt( (1/x) * serreverse( x*eta(x +x^2*O(x^n))^2/eta(x^2 +x^2*O(x^n))^4 ) ) ); polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^(n*(n+1)).
(2) A(x) = sqrt( (1/x) * Series_Reversion( x*eta(x)^2/eta(x^2)^4 ) ), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(3) A(x) = Product_{n>=1} (1 + (x*A(x)^2)^n) * (1 - (x*A(x)^2)^(2*n)).
(4) A(x) = exp( Sum_{n>=1} ( (x*A(x)^2)^n / (1 + (x*A(x)^2)^n) )/n ).
(5) A(x)^4 = Sum_{n>=0} (2*n+1) * (x*A(x)^2)^n / (1 - (x*A(x)^2)^(2*n+1)).
(6) A(x^2)^2 = Sum_{n>=0} (x*A(x^2))^n / (1 + (x*A(x^2))^(2*n+1)).
a(n) ~ c * d^n / n^(3/2), where d = 5.33733388876021052204016376282654316742329168165380444126... and c = 0.24712373554952847890961627688964866920906379264976188659... - Vaclav Kotesovec, Jun 13 2025

cp's OEIS Frontend

A106335 Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.

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A366174 Decimal expansion of a constant related to the asymptotics of A106336.

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A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

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A106507 G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).

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A106337 Number of ways of writing n as the sum of n triangular numbers.

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A106334 Decimal expansion of the function F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

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A106333 Decimal expansion of the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

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A106334 Decimal expansion of the function F(x) evaluated at the constant x that satisfies: F(x) - xF'(x) = 0, where F(x) = Sum_{n>=0} x^(n(n+1)/2).

A106333 Decimal expansion of the constant x that satisfies: F(x) - xF'(x) = 0, where F(x) = Sum_{n>=0} x^(n(n+1)/2).

A194042 G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n(n+1)/2) )^8 where q=xA(x).

A384829 G.f. satisfies A(x) = Sum_{n>=0} x^(n(n+1)/2) A(x)^(n*(n+1)).