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 1-5 of 5 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

Views

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

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.

A309045 Expansion of Product_{k>=0} (1 + x^(3^k) + x^(2*3^k) + x^(3^(k+1)))^(2^k).

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 5, 3, 3, 11, 8, 8, 19, 11, 11, 25, 14, 14, 41, 27, 27, 59, 32, 32, 70, 38, 38, 110, 72, 72, 158, 86, 86, 190, 104, 104, 289, 185, 185, 395, 210, 210, 455, 245, 245, 645, 400, 400, 829, 429, 429, 915, 486, 486, 1269, 783, 783, 1623, 840, 840, 1800, 960, 960, 2472
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 09 2019

Keywords

Comments

The trisection equals the self-convolution of this sequence.

Crossrefs

Cf. A054390, A237651, A309046.

Programs

  • Mathematica
    nmax = 63; CoefficientList[Series[Product[(1 + x^(3^k) + x^(2 3^k) + x^(3^(k + 1)))^(2^k), {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 63; A[] = 1; Do[A[x] = (1 + x + x^2 + x^3) A[x^3]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=0} ((1 - x^(4*3^k))/(1 - x^(3^k)))^(2^k).
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3) * A(x^3)^2.

A309046 Expansion of Product_{k>=0} (1 + x^(3^k) + x^(2*3^k) + x^(3^(k+1)))^(3^k).

Original entry on oeis.org

1, 1, 1, 4, 3, 3, 9, 6, 6, 25, 19, 19, 58, 39, 39, 105, 66, 66, 211, 145, 145, 394, 249, 249, 630, 381, 381, 1114, 733, 733, 1903, 1170, 1170, 2889, 1719, 1719, 4827, 3108, 3108, 7869, 4761, 4761, 11574, 6813, 6813, 18489, 11676, 11676, 28839, 17163, 17163, 41013, 23850
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 09 2019

Keywords

Comments

The trisection equals the three-fold convolution of this sequence with themselves.

Crossrefs

Cf. A054390, A237651, A309045, A321344.

Programs

  • Mathematica
    nmax = 52; CoefficientList[Series[Product[(1 + x^(3^k) + x^(2 3^k) + x^(3^(k + 1)))^(3^k), {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 52; A[] = 1; Do[A[x] = (1 + x + x^2 + x^3) A[x^3]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=0} ((1 - x^(4*3^k))/(1 - x^(3^k)))^(3^k).
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3) * A(x^3)^3.

A357366 Expansion of Product_{k>=0} 1 / (1 - x^(2^k) - x^(2^(k+1)))^(2^k).

Original entry on oeis.org

1, 1, 4, 5, 18, 23, 59, 82, 203, 285, 610, 895, 1838, 2733, 5217, 7950, 14763, 22713, 40526, 63239, 110652, 173891, 297529, 471420, 796706, 1268126, 2116508, 3384634, 5606444, 8991078, 14791302, 23782380, 38955441, 62737821, 102388280, 165126101, 268844542, 433970643
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2022

Keywords

Crossrefs

Cf. A073709, A084782, A173285, A237651.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)))^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 37; A[] = 1; Do[A[x] = A[x^2]^2/(1 - x - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = A(x^2)^2 / (1 - x - x^2).
a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=1} 1/(1 - x^(2^k) - x^(2^(k+1)))^(2^k) = 11.1991985012843182084779984477952870732899201240395056... and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 08 2022
Showing 1-5 of 5 results.

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