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.

A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2005

Keywords

Comments

a(n) = Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch, May 15 2008

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
The g.f. A = A(x) also satisfies the identities:
(1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
(2) A(x) = 1 + x*A/(1-x*A) + x^2*A^2/((1-x*A)*(1-x^2*A^2)) + x^3*A^3/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3)) + ...
(3) A(x) = 1 + x*A/(1-x*A)^2 + x^4*A^4/((1-x*A)*(1-x^2*A^2))^2 + x^9*A^9/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3))^2 + ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(2*(1-x^2*A(x)^2)) + x^3*A(x)^3/(3*(1-x^3*A(x)^3)) + x^4*A(x)^4/(4*(1-x^4*A(x)^4)) + x^5*A(x)^5/(5*(1-x^5*A(x)^5)) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 + 12405*x^7/7 + 62337*x^8/8 + ... + A008485(n)*x^n/n + ...

Links

Crossrefs

Cf. A109084, A184362, A008485, A000041; related: A106336.

Programs

  • Mathematica
    A109085list[n_] := Module[{m = 1, A = 1 + x}, For[i = 1, i <= n, i++, A = 1/Product[(1 - x^k*(A + x*O[x]^n)^k), {k, 1, n}]]; CoefficientList[A, x][[1 ;; n]]]; A109085list[24] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)
InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    {a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,n,x^m*A^m/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,sqrtint(n+1),(x*A)^(m^2)/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)^2))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(x^m*A^m/m)/(1-x^m*A^m+x*O(x^n)) )));polcoeff(A,n)} \\ Paul D. Hanna, Jun 01 2011
  • PARI
    {A008485(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A008485(m)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2012

Formula

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x*eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
(3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
(4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
(5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
(6) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^n/(1 - x^n*A(x)^n) ). - Paul D. Hanna, Jun 01 2011
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. - Paul D. Hanna, Feb 06 2012
a(n) = ([x^n] 1/((x; x)inf)^(n+1))/(n+1), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 25 2016

Keywords

Examples

			5.352701333486642687772415814165327879851483271286947097319690756...

Links

Crossrefs

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

Programs

  • Mathematica
    RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Formula

Equals limit n->infinity A008485(n)^(1/n).

A184362 G.f.: eta(x) + x*eta'(x).

Original entry on oeis.org

1, -2, -3, 0, 0, 6, 0, 8, 0, 0, 0, 0, -13, 0, 0, -16, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0, -36, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -71, 0, 0, 0, 0, 0, 0, -78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 93, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Jan 18 2011

Keywords

Comments

The formulas specified in this entry use eta(x) to denote Dedekind's eta(q) function without the q^(1/24) factor.

Examples

			G.f.: A(x) = 1 - 2*x - 3*x^2 + 6*x^5 + 8*x^7 - 13*x^12 - 16*x^15 + 23*x^22 + 27*x^26 - 36*x^35 - 41*x^40 +...
Illustrate the property: [x^n] A(x)/eta(x)^(n+1) = 0
in the table of coefficients of A(x)/eta(x)^(n+1) for n=0..10:
[1, -1, -3, -4, -7, -6, -12, -8, -15, -13, -18,...,-sigma(n),...];
[1,(0), -2, -6, -15, -28, -55, -90, -154, -240, -378,...];
[1, 1,(0), -5, -20, -54, -130, -275, -555, -1050, -1924,...];
[1, 2, 3,(0), -17, -72, -221, -572, -1350, -2958, -6160,...];
[1, 3, 7, 10,(0), -63, -287, -930, -2580, -6475, -15162,...];
[1, 4, 12, 26, 38,(0), -253, -1196, -4059, -11780, -31027,...];
[1, 5, 18, 49, 105, 153,(0), -1062, -5175, -18140, -54544,...];
[1, 6, 25, 80, 210, 442, 646,(0), -4615, -22990, -82671,...];
[1, 7, 33, 120, 363, 924, 1926, 2816,(0), -20570, -104285,...];
[1, 8, 42, 170, 575, 1668, 4161, 8602, 12585,(0), -93538,...];
[1, 9, 52, 231, 858, 2756, 7766, 19071, 39182, 57343,(0),...]; ...
so that the coefficient of x^n in A(x)/eta(x)^(n+1) is zero for n>=1.
Note: the g.f.s of the diagonals in the above table are powers of G(x),
where G(x) = 1/eta(x*G(x)) is the g.f. of A109085.
The g.f. of A184363 equals:
A(x)*eta(x)^2 = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...+ (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2) +...

Crossrefs

Cf. A184363, A184365, A109084, A109085, A000203.

Programs

  • PARI
    {a(n)=polcoeff(eta(x+x*O(x^n)) + x*deriv(eta(x+x*O(x^n))),n)}

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)/eta(x)^(n+1) = 0 for n>=1.
(2) [x^n] A(x)/eta(x)^n = A109084(n) for n>=0.
(3) [x^n] A(x)/eta(x)^(n+2) = A109085(n) for n>=0.
(4) A(x)/eta(x) = 1 - Sum_{n>=1} sigma(n)*x^n.
(5) A(x) = 1 + Sum_{n>=1} (-1)^n*(n*(3*n-1)/2+1 + (n*(3*n+1)/2+1)*x^n) * x^(n*(3*n-1)/2).
(6) A(x)*eta(x)^2 = Sum_{n>=0} (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2).

Extensions

Example of g.f. corrected by Paul D. Hanna, Jan 18 2011
Name changed slightly by Paul D. Hanna, Nov 27 2012
Showing 1-3 of 3 results.

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