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.

A183036 G.f.: exp( Sum_{n>=1} A001511(n)*2^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

Original entry on oeis.org

1, 2, 6, 10, 24, 38, 74, 110, 200, 290, 486, 682, 1096, 1510, 2314, 3118, 4650, 6182, 8946, 11710, 16616, 21522, 29886, 38250, 52328, 66406, 89394, 112382, 149496, 186610, 245086, 303562, 394814, 486066, 625686, 765306, 977112, 1188918, 1504954
Offset: 0

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Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

Compare to B(x), the g.f. of the binary partitions (A000123):
B(x) = exp( Sum_{n>=1} 2^A001511(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(2^n)).
2^A001511(n) exactly divides 2n.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 24*x^4 + 38*x^5 + 74*x^6 +...
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 24*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 64*x^8/8 + 2*x^9/9 + 8*x^10/10 +...+ A183037(n)*x^n/n +...

Crossrefs

Cf. A183037, A183038, A001511, A000123.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,valuation(2*m,2)*2^valuation(2*m,2)*x^m/m)+x*O(x^n)),n)}

Formula

G.f. satisfies: A(x) = (1-x^2)/(1-x)^2 * A(x^2)^2/A(x^4).

A183039 a(n) = A051064(n)*3^A051064(n) where A051064(n) equals the 3-adic valuation of 3n.

Original entry on oeis.org

3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 324, 3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 324, 3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 81, 3, 3, 18, 3, 3, 18, 3, 3, 1215, 3, 3, 18, 3, 3, 18, 3, 3, 81
Offset: 1

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

3^A051064(n) exactly divides 3n.

Examples

			L.g.f.: A(x) = 3*x + 3*x^2/2 + 18*x^3/3 + 3*x^4/4 + 3*x^5/5 + 18*x^6/6 + 3*x^7/7 + 3*x^8/8 + 81*x^9/9 + 3*x^10/10 + 3*x^11/11 + 18*x^12/12 +...
The g.f. of A183038 begins:
exp(A(x)) = 1 + 3*x + 6*x^2 + 15*x^3 + 30*x^4 + 51*x^5 + 93*x^6 +...

Links

Crossrefs

Cf. A183038, A183037.

Programs

  • PARI
    {a(n)=valuation(3*n,3)*3^valuation(3*n,3)}

Formula

Logarithmic derivative of A183038.

A195761 G.f.: exp( Sum_{n>=1} A055457(n) * 5^A055457(n) * x^n/n ) where 5^A055457(n) exactly divides 5*n.

Original entry on oeis.org

1, 5, 15, 35, 70, 135, 255, 465, 810, 1345, 2180, 3480, 5465, 8410, 12645, 18720, 27405, 39690, 56785, 80120, 111840, 154805, 212590, 289485, 390495, 522640, 694955, 918490, 1206310, 1573495, 2040260, 2631955, 3379065, 4317210, 5487145, 6941780, 8746180, 10977565, 13725310
Offset: 0

Views

Author

Paul D. Hanna, Sep 23 2011

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 135*x^5 + 255*x^6 +...
log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 50*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 50*x^10/10 +...
The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
Q0: [1, 135, 2180, 18720, 111840, 522640, 2040260, 6941780, ...];
Q1: [5, 255, 3480, 27405, 154805, 694955, 2631955, 8746180, ...];
Q2: [15, 465, 5465, 39690, 212590, 918490, 3379065, 10977565, ...];
Q3: [35, 810, 8410, 56785, 289485, 1206310, 4317210, 13725310, ...];
Q4: [70, 1345, 12645, 80120, 390495, 1573495, 5487145, 17090945, ...].

Links

Crossrefs

Cf. A195760, A183038, A183036, A055457.

Programs

  • PARI
    {a(n)=local(N=ceil(log(n+6)/log(5)));polcoeff(1/prod(k=0,N,(1-x^(5^k) +x*O(x^n))^(4*k+5)),n)}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, valuation(5*m, 5)*5^valuation(5*m, 5)*x^m/m)+x*O(x^n)), n)}

Formula

G.f.: A(x) = Product_{n>=0} 1/(1 - x^(5^n))^(4*n+5).
G.f. satisfies: A(x) = (1-x^5)/(1-x)^5 * A(x^5)^2/A(x^25).
G.f. satisfies: A(x) = A(x^5)*G(x) where G(x) = G(x^5)*(1-x^5)/(1-x)^5 is the g.f. of A195760.
Let the QUINTISECTIONS of A(x) be defined by:
A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
then
_ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
_ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
_ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
_ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
_ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
where R(x) = (1-x)^5*Product_{n>=0} (1 - x^(5^n))^(4*n+9).
Further, the quintisections are related by:
_ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.
Showing 1-3 of 3 results.

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