A370146 -id:A370146 - cp's OEIS frontend

A370145 Expansion of ( (1 + x)(1 - 2x)(1 + 4x) )^(1/3).

1, 1, -3, 3, -12, 30, -102, 318, -1083, 3657, -12747, 44715, -159222, 571332, -2068608, 7538664, -27646374, 101915850, -377496030, 1404077790, -5242135728, 19637862132, -73793090676, 278068062756, -1050503580534, 3977985415746, -15096209345958, 57403753019238, -218683959367908

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

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

			G.f.: A(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 - 159222*x^12 + ...
where A(x)^3 = (1 + 3*x - 6*x^2 - 8*x^3).
RELATED SERIES.
The following infinite product equals the g.f. of A370015:
A(x)^3 * A(2*x^2)^3 * A(4*x^3)^3 * A(8*x^4)^3 * ... * A(2^(n-1)*x^n)^3 * ... = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ... by the Jacobi triple product identity.
If A(x) = 1/B(x/A(x)) then B(x) = 1/A(x/B(x)) begins
B(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 + 431943435*x^32 - 8192222064*x^36 + ... + A370146(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630
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. A370015, A370016, A370146, A370147, A370149.

CoefficientList[Series[Surd[(1+x)(1-2x)(1+4x),3],{x,0,30}],x] (* Harvey P. Dale, Oct 04 2024 *)

{a(n) = polcoeff( (1 + 3*x - 6*x^2 - 8*x^3 +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 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3).
(2) Product_{n>=1} A( 2^(n-1)*x^n )^3 = Sum_{n>=0} 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2), which is the g.f. of A370015.
(3) A(x) = 1/B(x/A(x)) where B(x) = 1/A(x/B(x)) = x/Series_Reversion(x/A(x)) equals the g.f. of A370146.
a(n) ~ (-1)^(n+1) * 2^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 23 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.

cp's OEIS Frontend

A370145 Expansion of ( (1 + x)(1 - 2x)(1 + 4x) )^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

cp's OEIS Frontend

A370145 Expansion of ( (1 + x)*(1 - 2*x)*(1 + 4*x) )^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A370145 Expansion of ( (1 + x)(1 - 2x)(1 + 4x) )^(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.