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.

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A304318 O.g.f. A(x) satisfies: [x^n] exp( n*(n-1) * x ) / A(x) = 0.

Original entry on oeis.org

1, 0, 2, 24, 436, 10656, 328112, 12183456, 529242224, 26309617536, 1472135847072, 91526938123008, 6258004268952064, 466599240364076544, 37672137946943244288, 3274012281487011586560, 304724394621209905647360, 30239686358027369113804800, 3187164738879981461171955200, 355548230503664593634743375872
Offset: 0

Views

Author

Paul D. Hanna, May 11 2018

Keywords

Comments

It is striking that the coefficients of o.g.f. A(x) consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n-1)*x) / A(x) begins:
n=0: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=1: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];
n=2: [1, 2, 0, -160, -11600, -1376928, -250428416, -64479262720, ...];
n=3: [1, 6, 32, 0, -13392, -1630944, -286447104, -71981250048, ...];
n=4: [1, 12, 140, 1440, 0, -1916928, -351444096, -85338800640, ...];
n=5: [1, 20, 396, 7616, 128512, 0, -417488000, -107269127680, ...];
n=6: [1, 30, 896, 26496, 760752, 19101600, 0, -128348167680, ...];
n=7: [1, 42, 1760, 73440, 3034800, 121743072, 4260708864, 0, ...];
n=8: [1, 56, 3132, 174800, 9716608, 535021056, 28597069696, 1331047703552, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)*x ) / A(x) = 0 for n>=0.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304316:
A'(x)/A(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ... + A304316(n)*x^n + ...

Links

Crossrefs

Cf. A304316, A304319, A304320.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m-2) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(n - 1/2) / (sqrt(Pi) * c^(n - 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

A304317 Logarithmic derivative of F(x) that satisfies: [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

Original entry on oeis.org

2, 16, 260, 6200, 191832, 7235152, 320372320, 16243028896, 926219213216, 58608051937536, 4072302306624576, 308103163039452416, 25206121081157341184, 2216835666085110716928, 208552041718766935142400, 20896682922005650384144896, 2221700463749100463405564416, 249800738062720558095843241984, 29615243677328447562465854639104
Offset: 0

Views

Author

Paul D. Hanna, May 11 2018

Keywords

Examples

			O.g.f.: L(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304319:
F(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ... + A304319(n)*x^n + ...
which satisfies [x^n] exp( n*(n+1) * x ) / F(x) = 0 for n>0.

Links

Crossrefs

Cf. A304319, A304316, A304321.

Programs

  • Mathematica
    m = 25;
F = 1 + Sum[c[k] x^k, {k, m}];
s[n_] := Solve[SeriesCoefficient[Exp[n*(n + 1)*x]/F, {x, 0, n}] == 0][[1]];
Do[F = F /. s[n], {n, m}];
CoefficientList[D[F, x]/F + O[x]^m, x] (* Jean-François Alcover, May 21 2018 *)
  • PARI
    {a(n) = my(A=[1],L); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

Logarithmic derivative of the o.g.f. of A304319.
a(n) ~ sqrt(1-c) * 2^(2*n + 2) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 3/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020
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