cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A326606 G.f.: [ Sum_{n>=0} (2*n + 1) * x^n * (9 - x^n)^n ]^(1/3).

Original entry on oeis.org

1, 9, 53, 504, 3479, 34362, 248799, 2483091, 18383088, 185472450, 1378756330, 14142832704, 104308903182, 1093968797580, 7930694023345, 85563977678775, 604256899812240, 6755825768907204, 46021002487605408, 538061313239561853, 3494583984824812425, 43217684276354830458, 263780496112409697816, 3501503522404393600863, 19716335477199319610336
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2019

Keywords

Comments

G.f. is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n).
The cube of the g.f., A(x)^3, is congruent modulo 9 to Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)), and equals the g.f. of A326605.
First negative term is a(38) = -19995469991861952392493964610982114.

Examples

			G.f. A(x) = 1 + 9*x + 53*x^2 + 504*x^3 + 3479*x^4 + 34362*x^5 + 248799*x^6 + 2483091*x^7 + 18383088*x^8 + 185472450*x^9 + 1378756330*x^10 + ...
such that
A(x)^3 = 1 + 3*x*(9-x) + 5*x^2*(9-x^2)^2 + 7*x^3*(9-x^3)^3 + 9*x^4*(9-x^4)^4 + 11*x^5*(9-x^5)^5 + 13*x^6*(9-x^6)^6 + 15*x^7*(9-x^7)^7 + ...
Also,
A(x)^3 = (1 + 9*x)/(1 - 9*x)^2 - (3 + 9*x^2)*x^2/(1 - 9*x^2)^3 + (5 + 9*x^3)*x^6/(1 - 9*x^3)^4 - (7 + 9*x^4)*x^12/(1 - 9*x^4)^5 + (9 + 9*x^5)*x^20/(1 - 9*x^5)^6 - (11 + 9*x^6)*x^30/(1 - 9*x^6)^7 + (13 + 9*x^7)*x^42/(1 - 9*x^7)^8 + ...
RELATED SERIES.
G.f. A(x) is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n),
A(x) (mod 3) = 1 - x^2 - x^4 + x^10 + x^14 - x^24 - x^30 + x^44 + x^52 - x^70 - x^80 + x^102 + x^114 - x^140 - x^154 + x^184 + x^200 + ...
The cube of the g.f. A(x) begins
A(x)^3 = 1 + 27*x + 402*x^2 + 5103*x^3 + 58959*x^4 + 649539*x^5 + 6907037*x^6 + 71744535*x^7 + 731768013*x^8 + 7360989480*x^9 + ... + A326605(n)*x^n + ...
which is congruent modulo 9 to
A(x)^3 (mod 9) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 - 15*x^56 + 17*x^72 - 19*x^90 + 21*x^110 + ... + (-1)^n*(2*n+1)*x^(n*(n+1)) + ...

Links

Crossrefs

Cf. A326605.

Programs

  • PARI
    /* By definition */
{a(n) = my(A = sum(m=0,n, (2*m + 1) * x^m * (9 - x^m + x*O(x^n))^m)^(1/3) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    /* Accelerated series */
{a(n) = my(A = sum(m=0,sqrtint(n+1), (-1)^m * (2*m + 1 + 9*x^(m+1))* x^(m*(m+1)) / (1 - 9*x^(m+1) + x*O(x^n))^(m+2) )^(1/3) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f.: [ Sum_{n>=0} (2*n+1) * x^n * (9 - x^n)^n ]^(1/3).
G.f.: [ Sum_{n>=0} (-1)^n * (2*n+1 + 9*x^(n+1)) * x^(n*(n+1)) / (1 - 9*x^(n+1))^(n+2) ]^(1/3).

A326607 G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.

Original entry on oeis.org

1, 3, 2, 7, -1, 11, -3, 15, -19, 40, -34, 23, -6, 27, -76, 141, -139, 35, 61, 39, -270, 358, -208, 47, -35, 106, -298, 739, -874, 59, 725, 63, -1415, 1332, -526, 596, -617, 75, -664, 2185, -2069, 83, 1519, 87, -4696, 5740, -988, 95, -3571, 204, 3236, 4863, -8995, 107, 1891, 7701, -11578, 6784, -1594, 119, -941
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2019

Keywords

Comments

Compare the g.f. to the series: Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)) = Product_{n>=1} (1 - x^(2*n))^3.
Compare the g.f. to the series: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Examples

			G.f.: A(x) = 1 + 3*x + 2*x^2 + 7*x^3 - x^4 + 11*x^5 - 3*x^6 + 15*x^7 - 19*x^8 + 40*x^9 - 34*x^10 + 23*x^11 - 6*x^12 + 27*x^13 - 76*x^14 + 141*x^15 + ...
where A(x) is equal to the series given by
A(x) = 1 + 3*x*(1-x) + 5*x^2*(1-x^2)^2 + 7*x^3*(1-x^3)^3 + 9*x^4*(1-x^4)^4 + 11*x^5*(1-x^5)^5 + 13*x^6*(1-x^6)^6 + 15*x^7*(1-x^7)^7 + ...
Also,
A(x) = (1 + x)/(1 - x)^2 - (3 + x^2)*x^2/(1 - x^2)^3 + (5 + x^3)*x^6/(1 - x^3)^4 - (7 + x^4)*x^12/(1 - x^4)^5 + (9 + x^5)*x^20/(1 - x^5)^6 - (11 + x^6)*x^30/(1 - x^6)^7 + (13 + x^7)*x^42/(1 - x^7)^8 + ...

Links

Crossrefs

Cf. A326605, A260180.

Programs

  • PARI
    /* By definition */
{a(n) = my(A = sum(m=0, n, (2*m + 1) * x^m * (1 - x^m + x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    /* Accelerated series */
{a(n) = my(A = sum(m=0, sqrtint(n+1), (-1)^m * (2*m + 1 + x^(m+1))* x^(m*(m+1)) / (1 - x^(m+1) + x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2*n+1) * x^n * (1 - x^n)^n.
G.f.: Sum_{n>=0} (-1)^n * (2*n+1 + x^(n+1)) * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
Showing 1-2 of 2 results.

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