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.

A194043 G.f. satisfies: A(x) = Sum_{n>=0} q^(n*(n+1)/2) where q = x*A(x)^8.

Original entry on oeis.org

1, 1, 8, 93, 1272, 19058, 302705, 5007234, 85341048, 1488247757, 26428516221, 476280318599, 8688447566960, 160129594727162, 2977120816382182, 55769513231690143, 1051605638580341912, 19944384212454080598, 380202253988763211734, 7281079151205266545359
Offset: 0

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Author

Paul D. Hanna, Aug 12 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 +...
Let q = x*A(x)^8, then g.f. A(x) satisfies:
(0) A(x) = 1 + q + q^3 + q^6 + q^10 + q^15 + q^21 + q^28 + q^36 +...
The g.f. A(x) also satisfies:
(1) A(x)^2 = A(x) + x*A(x)^9 + x^3*A(x)^25 + x^6*A(x)^49 + x^10*A(x)^81 + x^15*A(x)^121 + x^21*A(x)^169 + x^28*A(x)^225 +...
(2) A(x)^4 = 1/(1-x*A(x)^8) + 3*x*A(x)^8/(1-x^3*A(x)^24) + 5*x^2*A(x)^16/(1-x^5*A(x)^40) + 7*x^3*A(x)^24/(1-x^7*A(x)^56) +...
(3) A(x)^8 = 1/(1-x^2*A(x)^16) + 8*x*A(x)^8/(1-x^4*A(x)^32) + 27*x^2*A(x)^16/(1-x^6*A(x)^48) + 64*x^3*A(x)^24/(1-x^8*A(x)^64) +...
(4) A(x) = (1+x*A(x)^8)*(1-x^2*A(x)^16) * (1+x^2*A(x)^16)*(1-x^4*A(x)^32) * (1+x^3*A(x)^24)*(1-x^6*A(x)^48) * (1+x^4*A(x)^32)*(1-x^8*A(x)^64) *...
(5) log(A(x)) = x*A(x)^8/(1+x*A(x)^8) + (x^2*A(x)^16/(1+x^2*A(x)^16))/2  + (x^3*A(x)^24/(1+x^3*A(x)^24))/3 + (x^4*A(x)^32/(1+x^4*A(x)^32))/4 +...
Related expansions begin:
_ A(x)^2 = 1 + 2*x + 17*x^2 + 202*x^3 + 2794*x^4 + 42148*x^5 + 672527*x^6 + 11161398*x^7 + 190702616*x^8 +...
_ A(x)^4 = 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 +...
_ A(x)^8 = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 + 4848456*x^6 + 82433472*x^7 + 1434755717*x^8 +...+ A194042(n)*x^n +...
_ log(A(x)) = x + 15*x^2/2 + 256*x^3/3 + 4619*x^4/4 + 85956*x^5/5 + 1631376*x^6/6 + 31387840*x^7/7 + 609993603*x^8/8 +...

Crossrefs

Cf. A194042, A194044.

Programs

  • PARI
    {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)^(1/8); polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,sqrt(2*n+1),x^(m*(m+1)/2)*(A+x*O(x^n))^((2*m+1)^2-1)));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*x^m*A^(8*m)/(1-x^(2*m+1)*(A+x*O(x^n))^(16*m+8)))^(1/4));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*x^m*A^(8*m)/(1-x^(2*m+2)*(A+x*O(x^n))^(16*m+16)))^(1/8));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1,n,A=prod(m=1,n,(1+(x*A^8)^m)*(1-(x*A^8)^(2*m)+x*O(x^n)))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1,n,A=exp(sum(m=1,n,(x*A^8)^m/(1+(x*A^8)^m+x*O(x^n))/m))); polcoeff(A, n)}

Formula

The g.f. A(x) satisfies:
(1) A(x)^2 = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^((2*n+1)^2),
(2) A(x)^4 = Sum_{n>=0} (2*n+1)*x^n*A(x)^(8*n)/(1 - x^(2*n+1)*A(x)^(16*n+8)),
(3) A(x)^8 = Sum_{n>=0} (n+1)^3*x^n*A(x)^(8*n)/(1 - x^(2*n+2)*A(x)^(16*n+16)),
(4) A(x) = Product_{n>=1} (1 + x^n*A(x)^(8*n))*(1 - x^(2*n)*A(x)^(16*n)),
(5) A(x) = exp( Sum_{n>=1} (x^n*A(x)^(8*n)/(1 + x^n*A(x)^(8*n)))/n ),
(6) A(x/F(x)^8) = F(x) where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
due to q-series identities.
Self-convolution 8th power equals A194042.
Self-convolution 4th power equals A194044.

A194044 G.f. satisfies: A(x) = ( Sum_{n>=0} q^(n*(n+1)/2) )^4 where q=x*A(x)^2.

Original entry on oeis.org

1, 4, 38, 472, 6685, 102340, 1649446, 27574712, 473750970, 8313682000, 148383186924, 2685209034144, 49154880453275, 908609423877476, 16935710715505290, 317951375873760120, 6006975695929624776, 114120962913881862036, 2178813296304338702764
Offset: 0

Views

Author

Paul D. Hanna, Aug 12 2011

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 +...
where
(0) A(x)^(1/4) = 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^21*A(x)^42 +... +...
(1) A(x) = 1/(1-x*A(x)^2) + 3*x*A(x)^2/(1-x^3*A(x)^6) + 5*x^2*A(x)^4/(1-x^5*A(x)^10) + 7*x^3*A(x)^6/(1-x^7*A(x)^14) +...
(2) A(x)^2 = 1/(1-x^2*A(x)^4) + 8*x*A(x)^2/(1-x^4*A(x)^8) + 27*x^2*A(x)^4/(1-x^6*A(x)^12) + 64*x^3*A(x)^6/(1-x^8*A(x)^16) +...
(3) A(x) = (1+x*A(x)^2)^4*(1-x^2*A(x)^4)^4 * (1+x^2*A(x)^4)^4*(1-x^4*A(x)^8)^4 * (1+x^3*A(x)^6)^4*(1-x^6*A(x)^12)^4 * (1+x^4*A(x)^8)^4*(1-x^8*A(x)^16)^4 *...
(4) log(A(x)) = 4*x*A(x)^2/(1+x*A(x)^2) + 4*(x^2*A(x)^4/(1+x^2*A(x)^4))/2  + 4*(x^3*A(x)^6/(1+x^3*A(x)^6))/3 + 4*(x^4*A(x)^8/(1+x^4*A(x)^8))/4 +...
Related expansions begin:
_ A(x)^(1/4) = 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)^2 = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 + 4848456*x^6 + 82433472*x^7 + 1434755717*x^8 +...+ A194042(n)*x^n +...

Crossrefs

Cf. A194042, A194043.

Programs

  • PARI
    {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)^(1/2); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(x*A^2+x*O(x^n))^(m*(m+1)/2))^4);polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(2*m+1)*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+1))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^3*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+2)))^(1/2));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=prod(m=1,n,(1+(x*A^2)^m)*(1-(x*A^2)^(2*m)+x*O(x^n)))^4); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,4*(x*A^2)^m/(1+(x*A^2)^m+x*O(x^n))/m))); polcoeff(A, n)}

Formula

The g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} (2*n+1)*x^n*A(x)^(2*n)/(1 - x^(2*n+1)*A(x)^(4*n+2)),
(2) A(x)^2 = Sum_{n>=0} (n+1)^3*x^n*A(x)^(2*n)/(1 - x^(2*n+2)*A(x)^(4*n+4)),
(3) A(x) = Product_{n>=1} (1 + x^n*A(x)^(2*n))^4*(1 - x^(2*n)*A(x)^(4*n))^4,
(4) A(x) = exp( Sum_{n>=1} 4*(x^n*A(x)^(2*n)/(1 + x^n*A(x)^(2*n)))/n ),
(5) A(x/F(x)^8) = F(x)^4 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
due to q-series identities.
Self-convolution 2nd power equals A194042.
Self-convolution 4th root equals A194043.
Showing 1-2 of 2 results.

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