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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A304319 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * x ) / A(x) = 0 for n>0.

Original entry on oeis.org

1, 2, 10, 104, 1772, 42408, 1303504, 48736000, 2139552016, 107629121888, 6094743943584, 383305860004992, 26491391713168640, 1994924925169038848, 162537118868301414912, 14243360542620058589184, 1335710880923054761115904, 133461369304858515494530560, 14154134380237986764584033792, 1587931951984022880659170662400
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.
Note that if [x^n] exp( (n+1)*(n+2)*x ) / G(x) = 0 then G(x) does not consist entirely of integer coefficients.

Examples

			O.g.f.: A(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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:
n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];
n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];
n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];
n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];
n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];
n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];
n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];
n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];
n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 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 A304317:
A'(x)/A(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 + ...+ A304317(n)*x^n + ...

Links

Crossrefs

Cf. A304317, A304318, A304320, A304863, A304864, A304865.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +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) * 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

A304316 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

4, 72, 1736, 53040, 1961728, 85062432, 4225904800, 236455369344, 14705880874944, 1005982098054912, 75048224139686912, 6062679436944758784, 527187725605767366144, 49092882744958427976192, 4874131922792403196021248, 513942386047796079510884352, 57356122407632751143615036416, 6754087907265415509369502427136, 836924235604443592471459956156416
Offset: 1

Views

Author

Paul D. Hanna, May 11 2018

Keywords

Examples

			O.g.f.: L(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 + ...
such that L(x) = F'(x)/F(x) where F(x) is the o.g.f. of A304324 :
F(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 + ... + A304318(n)*x^n + ...
which satisfies [x^n] exp( n*(n-1) * x ) / F(x) = 0 for n>0.

Links

Crossrefs

Cf. A304318, A304317, 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] // Rest (* 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-1)*(m-2) +x^2*O(x^m)) / Ser(A) )[m] ); L = Vec(Ser(A)'/Ser(A)); L[n]}
for(n=1,25, print1( a(n),", "))

Formula

Logarithmic derivative of the o.g.f. of A304318.
a(n) ~ sqrt(1-c) * 2^(2*n + 1) * n^(n + 3/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^(n+1) * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020
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