A370145 -id:A370145 - cp's OEIS frontend

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2) ]^(1/3).

1, 1, -1, 9, -18, 44, -54, 350, -1359, 3789, -9585, 42489, -163900, 543474, -1933092, 7499404, -27668718, 100329714, -371138346, 1394575578, -5236658316, 19587163968, -73536845444, 278088068628, -1052804678958, 3985553554074, -15132118280498, 57617112474306, -219680808219216

Offset: 0

Paul D. Hanna, Feb 22 2024

sign

Equals the self-convolution cube root of A370015.

			G.f.: A(x) = 1 + x - x^2 + 9*x^3 - 18*x^4 + 44*x^5 - 54*x^6 + 350*x^7 - 1359*x^8 + 3789*x^9 - 9585*x^10 + 42489*x^11 - 163900*x^12 + 543474*x^13 - 1933092*x^14 + 7499404*x^15 + ...
where the cube of g.f. A(x) yields the series
A(x)^3 = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + 11728213508096*x^36 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ...
The cube of g.f. A(x) also equals the infinite product
A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 2^2*x) * (1 + 2*x^2)*(1 - 2^2*x^2)*(1 + 2^3*x^2) * (1 + 2^2*x^3)*(1 - 2^3*x^3)*(1 + 2^4*x^3) * (1 + 2^3*x^4)*(1 - 2^4*x^4)*(1 + 2^5*x^4) * ...
Equivalently,
A(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) * (1 + 6*x^2 - 24*x^4 - 64*x^6)^(1/3) * (1 + 12*x^3 - 96*x^6 - 512*x^9)^(1/3) * (1 + 24*x^4 - 384*x^8 - 4096*x^12)^(1/3) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15)^(1/3) * ...
where
(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 + ... + A370145(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370015, A370145, A370148, A370334, A370019, A370336.

Cf. A304961, A370716, A370765.

Round[CoefficientList[Series[QPochhammer[-2, 2*x]^(1/3) * QPochhammer[-1/2, 2*x]^(1/3) * QPochhammer[2*x]^(1/3)*2^(1/3)/3^(2/3), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 23 2024 *)

{a(n) = my(A);
A = prod(m=1, n+1, (1 + 2^(m-1)*x^m) * (1 - 2^m*x^m) * (1 + 2^(m+1)*x^m) +x*O(x^n))^(1/3);
polcoeff(H=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)^3 = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + 2^(n-1)*x^n) * (1 - 2^n*x^n) * (1 + 2^(n+1)*x^n), by the Jacobi triple product identity.
(3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3).
a(n) ~ c * (-1)^(n+1) * 4^n / n^(4/3), where c = 0.260357663494676514371316... - Vaclav Kotesovec, Feb 23 2024
Radius of convergence r = 1/4 (from Vaclav Kotesovec's formula) and A(r) = ( Sum_{n>=0} (1/2^n + 2*2^n)/3 * 1/2^(n*(n+1)/2) )^(1/3) = ( Product_{n>=1} (1 + 1/2^n)*(1 - 1/4^(n+1)) )^(1/3) = 1.298389210904220681888354941631161162163... - Paul D. Hanna, Mar 07 2024

A370149 Expansion of ( (1 + x)(1 - 11x)(1 + 121x) )^(1/3).

Original entry on oeis.org

1, 37, -1776, 114096, -9165936, 810646320, -76152738288, 7450371782832, -750608233752432, 77319392827405872, -8104270335592602864, 861419406835986019248, -92621128795282877608560, 10055062260891607562940720, -1100545944769838408566122480, 121306087657061323164937678512

Offset: 0

Paul D. Hanna, Feb 25 2024

sign

The cube root of F(x) = (1 + x)*(1 - 11*x)*(1 + 121*x) = (1 + 111*x - 1221*x^2 - 1331*x^3) has integer coefficients because F(x) == (1+x)^3 (mod 9).

			G.f.: A(x) = 1 + 37*x - 1776*x^2 + 114096*x^3 - 9165936*x^4 + 810646320*x^5 - 76152738288*x^6 + 7450371782832*x^7 - 750608233752432*x^8 + ...
where A(x)^3 = (1 + 111*x - 1221*x^2 - 1331*x^3).
RELATED SERIES.
We have the following infinite product
A(x)^3 * A(11*x^2)^3 * A(11^2*x^3)^3 * A(11^3*x^4)^3 * ... = 1 + 111*x + 147631*x^3 + 2161452161*x^6 + 348104014265601*x^10 + 616687495357008127151*x^15 + ... + 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A370145, A370147, A370334.

{a(n) = polcoeff( ( (1 + x)*(1 - 11*x)*(1 + 121*x) +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 - 11*x)*(1 + 121*x) = (1 + 111*x - 1221*x^2 - 1331*x^3).
(2) Product_{n>=1} A( 11^(n-1)*x^n )^3 = Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 5^(1/3) * 11^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 25 2024

A370140 Expansion of g.f. A(x,y) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + yA(x,y)) = 1 + (y+1) Sum_{n>=1} x^(n(n+1)/2), as a triangle of coefficients T(n,k) of x^ny^k in A(x,y) read by rows.

Original entry on oeis.org

1, -1, -1, 1, 2, 1, -1, -3, -3, -1, 0, 3, 6, 4, 1, 1, 0, -7, -10, -5, -1, -1, -6, -1, 14, 15, 6, 1, 0, 12, 25, 3, -25, -21, -7, -1, 0, -14, -64, -75, -5, 41, 28, 8, 1, 1, 11, 102, 231, 179, 5, -63, -36, -9, -1, -1, -4, -109, -448, -651, -365, 0, 92, 45, 10, 1, 0, -5, 53, 593, 1486, 1546, 665, -14, -129, -55, -11, -1, 0, 12, 75, -407, -2342, -4077, -3241, -1114, 42, 175, 66, 12, 1

Offset: 1

Paul D. Hanna, Feb 12 2024

A370141(n) = Sum_{k=0..n-1} T(n,k), for n >= 1.
A370142(n) = Sum_{k=0..n-1} T(n,k) * 2^k, for n >= 1.
A370143(n) = Sum_{k=0..n-1} T(n,k) * 3^k, for n >= 1.
A370144(n) = Sum_{k=0..n-1} T(n,k) * 4^k, for n >= 1.

			G.f.: A(x,y) = x*(1) + x^2*(-1 - y) + x^3*(1 + 2*y + y^2) + x^4*(-1 - 3*y - 3*y^2 - y^3) + x^5*(3*y + 6*y^2 + 4*y^3 + y^4) + x^6*(1 - 7*y^2 - 10*y^3 - 5*y^4 - y^5) + x^7*(-1 - 6*y - y^2 + 14*y^3 + 15*y^4 + 6*y^5 + y^6) + x^8*(12*y + 25*y^2 + 3*y^3 - 25*y^4 - 21*y^5 - 7*y^6 - y^7) + x^9*(-14*y - 64*y^2 - 75*y^3 - 5*y^4 + 41*y^5 + 28*y^6 + 8*y^7 + y^8) + x^10*(1 + 11*y + 102*y^2 + 231*y^3 + 179*y^4 + 5*y^5 - 63*y^6 - 36*y^7 - 9*y^8 - y^9) + x^11*(-1 - 4*y - 109*y^2 - 448*y^3 - 651*y^4 - 365*y^5 + 92*y^7 + 45*y^8 + 10*y^9 + y^10) + ...
Let Q(x,y) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x,y) satisfies
(1) Q(x,y) = 1 + (x + y*A) + (x + y*A)*(x^2 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A)*(x^4 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A)*(x^4 + y*A)*(x^5 + y*A) + ...
also
(2) Q(x,y) = 1/(1 - y*A) + x/((1 - y*A)*(1 - x*y*A)) + x^3/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)) + x^6/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)) + x^10/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)*(1 - x^4*y*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x,y) = 1/(1 - (x + y*A)/(1 + x + y*A - (x^2 + y*A)/(1 + x^2 + y*A - (x^3 + y*A)/(1 + x^3 + y*A - (x^4 + y*A)/(1 + x^4 + y*A - (x^5 + y*A)/(1 + x^5 + y*A - (x^6 + y*A)/(1 + x^6 + y*A - (x^7 + y*A)/(1 - ...))))))))
where
Q(x,y) = 1 + (y+1)*x + (y+1)*x^3 + (y+1)*x^6 + (y+1)*x^10 + (y+1)*x^15 + (y+1)*x^21 + ... + (y+1)*x^(n*(n+1)/2) + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
 1;
 -1, -1;
 1, 2, 1;
 -1, -3, -3, -1;
 0, 3, 6, 4, 1;
 1, 0, -7, -10, -5, -1;
 -1, -6, -1, 14, 15, 6, 1;
 0, 12, 25, 3, -25, -21, -7, -1;
 0, -14, -64, -75, -5, 41, 28, 8, 1;
 1, 11, 102, 231, 179, 5, -63, -36, -9, -1;
 -1, -4, -109, -448, -651, -365, 0, 92, 45, 10, 1;
 0, -5, 53, 593, 1486, 1546, 665, -14, -129, -55, -11, -1;
 0, 12, 75, -407, -2342, -4077, -3241, -1114, 42, 175, 66, 12, 1;
 0, -15, -240, -391, 2087, 7481, 9736, 6182, 1749, -90, -231, -78, -13, -1;
 1, 19, 375, 1867, 1232, -8239, -20469, -20908, -10953, -2608, 165, 298, 91, 14, 1;
 -1, -26, -420, -3629, -9480, -2256, 26993, 49721, 41292, 18292, 3729, -275, -377, -105, -15, -1;
...
The g.f. of column 0 = (1-x) * Sum_{n>=1} x^(n*(n+1)/2).
The g.f. of column 1 = -(1-x)^2/(1+x) * [Sum_{n>=0} x^(n*(n+1)/2)] * [Sum_{n>=1} x^(n*(n+1)/2)]^2.

Paul D. Hanna, Table of n, a(n) for n = 1..2485

Cf. A370141 (y=1), A370142 (y=2), A370143 (y=3), A370144 (y=4).

Cf. A370145 (column 1), A370146 (column 2).

{T(n,k) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; polcoeff(A[n+1],k,y)}
for(n=1,21, for(k=0,n-1, print1(T(n,k),", "));print(""))

/* Generate A(x,y) recursively using integration wrt y */
{T(n,k) = my(A = x +x*O(x^n), Q = sum(m=1,sqrtint(2*n+1), x^(m*(m+1)/2) +x*O(x^n)) );
for(i=1,n, A = (1/y) * intformal( Q/sum(m=1,n, sum(j=1,m, prod(k=1,m, if(j==k,1, x^k + y*A) +O(x^n))) ), y) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,15,for(k=0,n-1, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following formulas.
Let Q(x,y) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x,y) = Sum_{n>=0} Product_{k=1..n} (x^k + y*A(x,y)).
(2) Q(x,y) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * y*A(x,y)).
(3) Q(x,y) = 1/(1 - F(1)), where F(n) = (x^n + y*A(x,y))/(1 + x^n + y*A(x,y) - F(n+1)), a continued fraction.
(4) A(x,0) = (1-x) * Sum_{n>=1} x^(n*(n+1)/2), which is the g.f. of column 0.
(5) d/dy A(x,y) at y = 0 equals -(1-x)^2/(1+x) * Q(x,0) * (Q(x,0) - 1)^2, which is the g.f. of column 1.

A370147 Expansion of ( (1 + x)(1 + 7x)(1 + 49x) )^(1/3).

Original entry on oeis.org

1, 19, -228, 6492, -216372, 7851180, -300848772, 11974587132, -490113592788, 20492868223308, -871404823013412, 37562003034015900, -1637401559515373172, 72053378865932154348, -3196217668534369463748, 142763786831538212246076, -6415201218873454789867284, 289797678008730755585589900

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

The cube root of F(x) = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).

In general, for k > 1, if g.f. = ((1 + x)*(1 + k*x)*(1 + k^2*x))^(1/3), then a(n) ~ (-1)^(n+1) * (k-1)^(2/3) * (k+1)^(1/3) * k^(2*n-1) / (3*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

			G.f.: A(x) = 1 + 19*x - 228*x^2 + 6492*x^3 - 216372*x^4 + 7851180*x^5 - 300848772*x^6 + 11974587132*x^7 - 490113592788*x^8 + ...
where A(x)^3 = (1 + 57*x + 399*x^2 + 343*x^3).
RELATED SERIES.
We have the following infinite product involving the g.f. A(x)
A(x)^3 * A(-7*x^2)^3 * A(49*x^3)^3 * A(-343*x^4)^3 * A(2401*x^5)^3 * ... = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 + ... + (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..406
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A370145, A370149, A370148.

{a(n) = polcoeff( ( (1 + x)*(1 + 7*x)*(1 + 49*x) +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3).
(2) Product_{n>=1} A( (-7)^(n-1)*x^n )^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 7^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

A370146 Expansion of x / Series_Reversion( x/(1 + 3x - 6x^2 - 8*x^3)^(1/3) ).

Original entry on oeis.org

1, -1, 3, 0, 0, 0, -9, 0, 27, 0, 0, 0, -324, 0, 1215, 0, 0, 0, -18711, 0, 75816, 0, 0, 0, -1301265, 0, 5484996, 0, 0, 0, -100048689, 0, 431943435, 0, 0, 0, -8192222064, 0, 35942240565, 0, 0, 0, -700434986472, 0, 3108770417700, 0, 0, 0, -61805774132388, 0, 276711654879477

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

The cube root of F(x) = (1 + 3*x - 6*x^2 - 8*x^3) = (1 + x)*(1 - 2*x)*(1 + 4*x) is an integer series because F(x) == (1+x)^3 (mod 9).

			G.f.: A(x) = = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + ...
RELATED SERIES.
If A(x) = 1/B(x/A(x)) then B(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) begins
B(x) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 - 12747*x^10 + 44715*x^11 + ... + A370145(n)*x^n + ...
A(x) = 1/D(x^6) + 3*x^2*D(x^6) - x, where
1/D(x) = 1 - 9*x - 324*x^2 - 18711*x^3 - 1301265*x^4 - 100048689*x^5 - 8192222064*x^6 - 700434986472*x^7 + ...
and D(x) = ( (1-sqrt(1-108*x))/(54*x) )^(1/3) begins
D(x) = 1 + 9*x + 405*x^2 + 25272*x^3 + 1828332*x^4 + 143981145*x^5 + 11980746855*x^6 + 1036256805900*x^7 + ... + 3^n*A008931(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370016, A370145, A008931, A000108.

{a(n) = polcoeff( x/serreverse( x/(1 + 3*x - 6*x^2 - 8*x^3 +x*O(x^n))^(1/3) ), n)}
for(n=0,50,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).
(2) A(x) = 1 / B(x/A(x)) where B(x) = 1 / A(x/B(x)) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) equals the g.f. of A370145.
(3) A(x) = 1/C(27*x^6)^(1/3) + 3*x^2*C(27*x^6)^(1/3) - x, where C(x) = 1 + x*C(x)^2  = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

A370779 Expansion of ( (1 + x)(1 - 5x)(1 + 25x) )^(1/3).

Original entry on oeis.org

1, 7, -84, 1020, -17220, 313068, -6075444, 122709468, -2553130020, 54317619660, -1175968479252, 25819593611196, -573476704909572, 12861006710141100, -290799326551852020, 6621725329384239516, -151707434284857934308, 3494405505576163607436

Offset: 0

Seiichi Manyama, Mar 01 2024

sign

The cube root of F(x) = (1 + x)*(1 - (3*k+2)*x)*(1 + (3*k+2)^2*x) is an integer series because F(x) == (1+x)*(1 + 2*x - 8*x^2) == (1+x)^3 (mod 9).

Cf. A370145, A370149, A370780.

Cf. A370782.

my(N=20, x='x+O('x^N)); Vec(((1+x)*(1-5*x)*(1+25*x))^(1/3))

A370780 Expansion of ( (1 + x)(1 - 17x)(1 + 289x) )^(1/3).

Original entry on oeis.org

1, 91, -9828, 1535868, -294731892, 62322050700, -13990450587012, 3270320252339868, -787131217405990548, 193694053976566000812, -48497295306135216560292, 12313491783703337923916220, -3162498663877264843739477172

Offset: 0

Seiichi Manyama, Mar 01 2024

sign

The cube root of F(x) = (1 + x)*(1 - (3*k+2)*x)*(1 + (3*k+2)^2*x) is an integer series because F(x) == (1+x)*(1 + 2*x - 8*x^2) == (1+x)^3 (mod 9).

Cf. A370145, A370149, A370779.

my(N=20, x='x+O('x^N)); Vec(((1+x)*(1-17*x)*(1+289*x))^(1/3))

A370781 Expansion of 1 / ( (1 - x)(1 + 2x)(1 - 4x) )^(1/3).

Original entry on oeis.org

1, 1, 4, 10, 37, 121, 442, 1576, 5818, 21466, 80272, 301324, 1138762, 4320226, 16459132, 62904664, 241134553, 926678569, 3569385772, 13776307714, 53267766997, 206304355225, 800203300354, 3108008802064, 12086612436376, 47056902019336, 183400211694496

Offset: 0

Seiichi Manyama, Mar 01 2024

nonn

Cf. A004987, A370145.

my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1+2*x)*(1-4*x))^(1/3))

a(n) = sum(k=0, n\2, (-9)^k*binomial(-1/3, k)*binomial(n, 2*k)); \\ Seiichi Manyama, Aug 18 2025

a(n) ~ 2^(2*n+1) / (Gamma(1/3) * 3^(2/3) * n^(2/3)). - Vaclav Kotesovec, Mar 10 2024

a(n) = Sum_{k=0..floor(n/2)} A004987(k) * binomial(n,2*k). - Seiichi Manyama, Aug 18 2025

cp's OEIS Frontend

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) ]^(1/3).

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A370149 Expansion of ( (1 + x)*(1 - 11*x)*(1 + 121*x) )^(1/3).

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A370140 Expansion of g.f. A(x,y) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + y*A(x,y)) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y) read by rows.

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A370147 Expansion of ( (1 + x)*(1 + 7*x)*(1 + 49*x) )^(1/3).

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A370146 Expansion of x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).

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A370779 Expansion of ( (1 + x)*(1 - 5*x)*(1 + 25*x) )^(1/3).

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A370780 Expansion of ( (1 + x)*(1 - 17*x)*(1 + 289*x) )^(1/3).

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A370781 Expansion of 1 / ( (1 - x)*(1 + 2*x)*(1 - 4*x) )^(1/3).

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A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2) ]^(1/3).

A370149 Expansion of ( (1 + x)(1 - 11x)(1 + 121x) )^(1/3).

A370140 Expansion of g.f. A(x,y) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + yA(x,y)) = 1 + (y+1) Sum_{n>=1} x^(n(n+1)/2), as a triangle of coefficients T(n,k) of x^ny^k in A(x,y) read by rows.

A370147 Expansion of ( (1 + x)(1 + 7x)(1 + 49x) )^(1/3).

A370146 Expansion of x / Series_Reversion( x/(1 + 3x - 6x^2 - 8*x^3)^(1/3) ).

A370779 Expansion of ( (1 + x)(1 - 5x)(1 + 25x) )^(1/3).

A370780 Expansion of ( (1 + x)(1 - 17x)(1 + 289x) )^(1/3).

A370781 Expansion of 1 / ( (1 - x)(1 + 2x)(1 - 4x) )^(1/3).