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 1-4 of 4 results.

A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 1, 4, 3, 15, 12, 37, 25, 100, 75, 219, 144, 501, 357, 972, 615, 1995, 1380, 3665, 2285, 7052, 4767, 12255, 7488, 22305, 14817, 37524, 22707, 65775, 43068, 106837, 63769, 180436, 116667, 286251, 169584, 471173, 301589, 729404, 427815, 1169211, 741396, 1778545, 1037149
Offset: 0

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Author

Paul D. Hanna, Sep 20 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...
where
log(A(x)) = x + 7*x^2/2 - 2*x^3/3 + 31*x^4/4 + x^5/5 - 14*x^6/6 + x^7/7 + 127*x^8/8 +...+ A195587(n)*x^n/n +...
Let C(x) be the odd bisection of g.f. A(x):
C(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 + 615*x^7 + 1380*x^8 + 2285*x^9 + 4767*x^10 + 7488*x^11 + 14817*x^12 +...+ A237650(n)*x^n +...
then C(x) equals the cube of an integer series:
C(x)^(1/3) = 1 + x + 3*x^2 + 2*x^3 + 9*x^4 + 7*x^5 + 17*x^6 + 10*x^7 + 41*x^8 + 31*x^9 + 75*x^10 + 44*x^11 + 150*x^12 +...+ A237651(n)*x^n +...
which equals A(x)/C(x^2)^(1/3).
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * (1+x^4+x^8)^6 * (1+x^8+x^16)^12 * (1+x^16+x^32)^24 *...* (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n) *...

Links

Crossrefs

Cf. A195587, A163658, A163659, A237650, A002487; variant: A237646.

Programs

  • PARI
    {A163659(n)=if(n<1,0,if(n%3,1,-2)*sigma(2^valuation(n,2)))}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    /* G.f.: A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2: */
{a(n)=local(A=1+x); for(i=1, #binary(n), A=(1+x+x^2)*(1+x^2+x^4)*subst(A^2, x, x^2) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* G.f.: (1+x+x^2) * Product_{n>=0} (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n): */
{a(n)=local(A=1+x); A=(1+x+x^2)*prod(k=0, #binary(n), (1+x^(2*2^k)+x^(4*2^k)+x*O(x^n))^(3*2^k)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A195587(n)*x^n/n ), where A195587(n) = A163659(n^2).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(3*2^n).
(3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
Bisections: let A(x) = B(x^2) + x*C(x^2), then
(4) B(x) = (1+x) * C(x).
(5) C(x) = (1+x+x^2)^3 * C(x^2)^2.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^3 = C(x) * C(x^2).
(8) A(x)^2 = C(x) / (1+x+x^2).
(9) A(x) = ( C(x)/A(x) - C(x^2)^2/A(x^2)^2 ) / (2*x).

Extensions

Entry and formulas revised by Paul D. Hanna, May 04 2014

A237649 a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 8191, 1, -30, 1, 127, -2, 15, 1, -2046, 1, 15, -2, 127, 1, -30, 1, 65535, -2, 15, 1, -254, 1, 15, -2, 1023, 1, -30, 1, 127, -2, 15, 1, -16382, 1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 524287, 1, -30, 1, 127
Offset: 1

Views

Author

Paul D. Hanna, May 03 2014

Keywords

Comments

Multiplicative because A163659 is. - Andrew Howroyd, Jul 27 2018

Examples

			L.g.f.: L(x) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A163659(n^3)*x^n/n +...
where
exp(L(x)) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 +...+ A237646(n)*x^n +...

Links

Crossrefs

Cf. A237646, A163659.

Programs

  • PARI
    {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=A163659(n^3)}
for(n=1,64,print1(a(n),", "))
  • PARI
    {a(n)=local(X=x+x*O(x^n),A);A=log(1+X+X^2) + sum(k=0,#binary(n),7*4^k*log(1 + X^(2*2^k) + X^(4*2^k)));n*polcoeff(A,n)}
for(n=1,64,print1(a(n),", "))

Formula

L.g.f.: log(1+x+x^2) + Sum_{n>=0} 7*4^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n.
G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 14*8^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)).

A237647 G.f. satisfies: A(x) = (1 + x + x^2)^7 * A(x^2)^4.

Original entry on oeis.org

1, 7, 56, 273, 1463, 6048, 26537, 97903, 377384, 1281497, 4502463, 14322560, 46849089, 141332583, 436556440, 1259742225, 3710541975, 10308494560, 29165172617, 78396244591, 214217633672, 559335671353, 1482519853311, 3772127020032, 9731443674113, 24191903115079, 60918829766648
Offset: 0

Views

Author

Paul D. Hanna, May 04 2014

Keywords

Examples

			G.f.: A(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 +...
where:
A(x) = (1+x+x^2)^7 * (1+x^2+x^4)^28 * (1+x^4+x^8)^112 * (1+x^8+x^16)^448 * (1+x^16+x^32)^896 *...* (1 + x^(2^n) + x^(2*2^n))^(7*4^n) *...

Crossrefs

Cf. A237646, A237648, A237650.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)^7*subst(A^4,x,x^2) +x*O(x^n));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);A=prod(k=0,#binary(n),(1+x^(2^k)+x^(2*2^k)+x*O(x^n))^(7*4^k));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

Formula

The odd-indexed bisection of A237646.
The 7th self-convolution of A237648.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(7*4^n).
(2) A(x) / A(-x) = (1+x+x^2)^7 / (1-x+x^2)^7.

A237648 G.f. satisfies: A(x) = (1 + x + x^2) * A(x^2)^4.

Original entry on oeis.org

1, 1, 5, 4, 30, 26, 106, 80, 459, 379, 1451, 1072, 5210, 4138, 14894, 10756, 47617, 36861, 127949, 91088, 376264, 285176, 957336, 672160, 2640964, 1968804, 6452260, 4483456, 16921416, 12437960, 39873688, 27435728, 100259070, 72823342, 229410006, 156586664, 556880812, 400294148
Offset: 0

Views

Author

Paul D. Hanna, May 03 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 +...
such that A(x) = (1+x+x^2) * A(x^2)^4, where:
A(x)^4 = 1 + 4*x + 26*x^2 + 80*x^3 + 379*x^4 + 1072*x^5 + 4138*x^6 + 10756*x^7 +...
The g.f. may thus be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^4 * (1+x^4+x^8)^16 * (1+x^8+x^16)^64 *...
Note that x*A(x^2)^7 is the odd bisection of the g.f. G(x) of A237646:
A(x)^7 = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 +...+ A237647(n)*x^n +...
G(x) = (1+x+x^2)*A(x^2)^7 = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 + 1463*x^9 + 7511*x^10 + 6048*x^11 +...+ A237646(n)*x^n +...

Links

Crossrefs

Cf. A237646, A237647.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)*subst(A^4,x,x^2) +x*O(x^n));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);A=prod(k=0,#binary(n),(1+x^(2^k)+x^(2*2^k)+x*O(x^n))^(4^k));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

Formula

The 7th self-convolution yields A237647.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(4^n).
(2) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
Bisections: let A(x) = B(x^2) + x*C(x^2), then
(3) B(x) = (1+x) * C(x).
(4) C(x) = A(x)^4 = (1+x+x^2)^4 * C(x^2)^4.
Showing 1-4 of 4 results.

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