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 1-3 of 3 results.

A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)*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, 8, 7, 63, 56, 329, 273, 1736, 1463, 7511, 6048, 32585, 26537, 124440, 97903, 475287, 377384, 1658881, 1281497, 5783960, 4502463, 18825023, 14322560, 61171649, 46849089, 188181672, 141332583, 577889023, 436556440, 1696298665, 1259742225, 4970284200, 3710541975, 14019036535, 10308494560
Offset: 0

Views

Author

Paul D. Hanna, May 03 2014

Keywords

Comments

Compare to the g.f. of A195586.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +...
where
log(A(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 +...+ A237649(n)*x^n/n +...
Bisections: let A(x) = B(x^2) + x*C(x^2), then:
B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +...
C(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 + 377384*x^8 + 1281497*x^9 + 4502463*x^10 +...+ A237647(n)*x^n +...
Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series:
C(x)^(1/7) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 + 379*x^9 + 1451*x^10 + 1072*x^11 + 5210*x^12 +...+ A237648(n)*x^n +...
Also, C(x) / (1+x+x^2)^3 = A(x)^4:
A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +...
Further, C(x)*C(x^2)^3 = A(x)^7:
A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +...
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^7 * (1+x^4+x^8)^28 * (1+x^8+x^16)^112 * (1+x^16+x^32)^448 *...* (1 + x^(2*2^n) + x^(4*2^n))^(7*4^n) *...

Crossrefs

Cf. A237647, A237648, A237649, A195586.

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^3)*x^k/k)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^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)^7 * C(x^2)^4.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^7 = C(x) * C(x^2)^3.
(8) A(x)^4 = C(x) / (1+x+x^2)^3.
(9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5).

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.

A237650 G.f. satisfies: A(x) = (1+x+x^2)^3 * A(x^2)^2.

Original entry on oeis.org

1, 3, 12, 25, 75, 144, 357, 615, 1380, 2285, 4767, 7488, 14817, 22707, 43068, 63769, 116667, 169584, 301589, 427815, 741396, 1037149, 1761087, 2418432, 4025153, 5465955, 8956716, 11986009, 19330347, 25633296, 40835973, 53508711, 84129156, 109392269, 170278047, 219206976
Offset: 0

Views

Author

Paul D. Hanna, May 04 2014

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 +...
where:
A(x) = (1+x+x^2)^3 * (1+x^2+x^4)^6 * (1+x^4+x^8)^12 * (1+x^8+x^16)^24 * (1+x^16+x^32)^48 *...* (1 + x^(2^n) + x^(2*2^n))^(3*2^n) *...

Crossrefs

Cf. A195586, A237651, A237647.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)^3*subst(A^2,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))^(3*2^k));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

Formula

The odd-indexed bisection of A195586.
The 3rd self-convolution of A237651.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(3*2^n).
(2) A(x) / A(-x) = (1+x+x^2)^3 / (1-x+x^2)^3.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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