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 1-6 of 6 results.

A129374 G.f. satisfies: A(x) = 1/(1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 20, 35, 48, 76, 103, 166, 221, 333, 451, 671, 894, 1303, 1730, 2479, 3288, 4615, 6086, 8502, 11142, 15299, 20034, 27285, 35514, 47937, 62168, 83259, 107650, 142929, 184090, 243207, 312041, 409210, 523709, 683261, 871239, 1130703
Offset: 0

Views

Author

Paul D. Hanna, Apr 12 2007

Keywords

Links

Crossrefs

Cf. A074206, A129373, A129375.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=2,n,A=1/(1-x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

Formula

G.f.: A(x) = Product_{n>=1} 1/(1 - x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.
a(n) ~ exp((1 + 1/r) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(10*(1+r))) / ((2*Pi)^(29/50) * sqrt(1+r) * n^((6 + 5*r)/(10*(1+r)))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A129373 G.f. satisfies: A(x) = (1+x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*...

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 9, 13, 19, 26, 34, 52, 67, 89, 123, 166, 214, 295, 380, 501, 660, 858, 1098, 1461, 1858, 2384, 3072, 3940, 4975, 6410, 8070, 10234, 12946, 16322, 20412, 25848, 32201, 40261, 50287, 62728, 77681, 96885, 119673, 148197, 183108, 225974
Offset: 0

Views

Author

Paul D. Hanna, Apr 12 2007

Keywords

Links

Crossrefs

Cf. A074206; A129374, A129375.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=2,n,A=(1+x)*prod(n=2,i,subst(A,x,x^n+x*O(x^i)))); polcoeff(A,n)}

Formula

G.f.: A(x) = Product_{n>=1} (1 + x^n)^A074206(n) where A074206(n) equals the number of ordered factorizations of n.
a(n) ~ exp((1 + 1/r) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(1+r)) * n^(r/(1+r))) * (-(1 - 2^(-r)) * Gamma(1+r) * Zeta(1+r) / Zeta'(r))^(1/(2 + 2*r)) / (2^(1/10) * sqrt(Pi) * sqrt(1+r) * n^((2+r)/(2 + 2*r))), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Nov 04 2018

A247667 Decimal expansion of the coefficient c_m in c_m*log(N), the asymptotic mean number of factors in a random factorization of n <= N.

Original entry on oeis.org

5, 5, 0, 0, 1, 0, 0, 0, 5, 4, 1, 3, 1, 5, 4, 4, 9, 1, 8, 3, 3, 0, 5, 8, 1, 2, 6, 7, 0, 2, 2, 2, 2, 1, 9, 6, 4, 6, 1, 1, 6, 8, 2, 2, 7, 1, 0, 2, 7, 1, 4, 0, 4, 0, 9, 8, 8, 8, 3, 9, 6, 5, 8, 5, 8, 9, 2, 9, 0, 5, 3, 0, 6, 6, 6, 6, 0, 5, 6, 4, 8, 5, 9, 5, 1, 1, 8, 7, 2, 0, 6, 5, 2, 3, 5, 3, 4, 6, 6, 5, 4
Offset: 0

Views

Author

Jean-François Alcover, Sep 22 2014

Keywords

Examples

			0.55001000541315449183305812670222219646116822710271404...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 293.

Links

Crossrefs

Cf. A107311, A129373, A129374, A129375, A217598, A247668.

Programs

  • Mathematica
    digits = 101; rho = x /. FindRoot[Zeta[x] == 2, {x, 2}, WorkingPrecision -> digits+5]; cm = -1/Zeta'[rho]; RealDigits[cm, 10, digits] // First

Formula

c_m = -1/zeta'(rho), where rho = 1.728647... is A107311, the real solution to zeta(rho) = 2.
Residue_{s = rho} 1/(2 - Zeta(s)). - Vaclav Kotesovec, Nov 04 2018

A307615 E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, 1, 5, 31, 337, 2741, 40621, 474475, 8461601, 132034537, 2648537461, 50079699671, 1204884343345, 26450428964701, 697107087763997, 17873985363570211, 526080367468142401, 15060611189639187665, 487251625325328212581, 15494976568071805188367, 545902629556769672596241
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 337*x^4/4! + 2741*x^5/5! + 40621*x^6/6! + 474475*x^7/7! + 8461601*x^8/8! + 132034537*x^9/9! + ...

Links

Crossrefs

Cf. A050369, A074206, A129375, A307604.

Programs

  • Mathematica
    terms = 20; A[] = 1; Do[A[x] = Exp[x] Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f.: exp(Sum_{k>=1} A050369(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} A074206(k)*k*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ (-Gamma(2+r)/zeta'(r))^(1/(4 + 2*r)) * exp(-n + 12/25 + n^(1 - 1/(2+r)) * (2+r) * (-Gamma(2+r)/zeta'(r))^(1/(2+r)) / (1+r)) * n^(n - 1/(4 + 2*r)) / sqrt(2+r), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Aug 09 2021

A307660 E.g.f. A(x) satisfies: A(x) = exp(-x) * A(x^2)*A(x^3)*A(x^4)* ... *A(x^k)* ...

Original entry on oeis.org

1, -1, -1, -1, -23, 139, -929, 12011, -54319, 664343, 7497631, 17751799, -1294263431, 13183537379, 335384855807, -8293330879261, 26192873684641, -1587651616174289, 12035003736999871, -887536237005983377, 13114291271436277001, -332542758207041951941, 2683832751567973018399
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

Examples

			E.g.f.: A(x) = 1 - x - x^2/2! - x^3/3! - 23*x^4/4! + 139*x^5/5! - 929*x^6/6! + 12011*x^7/7! - 54319*x^8/! + 664343*x^9/9! + ...

Crossrefs

Cf. A074206, A129375, A307661.

Programs

  • Mathematica
    terms = 22; A[] = 1; Do[A[x] = Exp[-x] Product[A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f.: exp(-Sum_{n>=1} A074206(k)*x^k).
a(0) = 1; a(n) = -Sum_{k=1..n} A074206(k)*k!*binomial(n-1,k-1)*a(n-k).

A385635 G.f. satisfies A(x) = x + Product_{n>=2} A(x^n) with A(0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 13, 15, 26, 26, 41, 48, 73, 80, 119, 136, 198, 225, 313, 367, 518, 585, 797, 941, 1264, 1466, 1953, 2285, 3022, 3524, 4571, 5391, 6993, 8152, 10440, 12316, 15684, 18370, 23236, 27327, 34389, 40364, 50370, 59292, 73880, 86547, 107080, 125976, 155266, 182058
Offset: 0

Views

Author

Paul D. Hanna, Jul 05 2025

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 8*x^8 + 8*x^9 + 13*x^10 + 15*x^11 + 26*x^12 + ...
where
A(x) = x + A(x^2)*A(x^3)*A(x^4)*A(x^5)* ... * A(x^n) * ...

Links

Crossrefs

Cf. A129374, A129375.

Programs

  • PARI
    {a(n) = my(A=1+x +x*O(x^n)); for(i=1, ceil(log(n+2)/log(2)), A = x + prod(k=2,#A,subst(A, x, x^k)) +x*O(x^n); ); polcoef(A, n)}
for(n=0, 50, print1(a(n), ", "))
Showing 1-6 of 6 results.

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