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.

A219268 Logarithmic derivative of A001142, where A001142(n) = product{k=1..n} k^k/k!.

Original entry on oeis.org

1, 3, 22, 347, 11986, 956334, 184142134, 87903876147, 105736320973732, 323943204887363938, 2547547949361933790328, 51735228018482706470521574, 2726127372514537039881847535054, 374214400937086673452020875815709240, 134262616041282033840675468757467513112522
Offset: 1

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

A001142(n) = hyperfactorial(n)/superfactorial(n) = A002109(n)/A000178(n).

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 22*x^3/3 + 347*x^4/4 + 11986*x^5/5 + 956334*x^6/6 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 9*x^3 + 96*x^4 + 2500*x^5 + 162000*x^6 + 26471025*x^7 + 11014635520*x^8 +...+ A001142(n)*x^n +...

Crossrefs

Cf. A001142, A219266, A219268.

Programs

  • Mathematica
    nmax=15; Rest[CoefficientList[Series[Log[Sum[Product[j^j/j!,{j,1,k}]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)
  • PARI
    {a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j/j!)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 - 2/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A219269 G.f. satisfies: A(x) = Sum_{n>=0} x^n/A(x)^n * Product_{k=0..n} k!.

Original entry on oeis.org

1, 1, 1, 8, 247, 33184, 24678266, 125237615376, 5055581949347115, 1834887966372111613136, 6658588234946979374670842054, 265790194051800257952649093995518288, 127313960109916568757252293587045497552163302, 792786695940715289991550398242378268738388375150573312
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 247*x^4 + 33184*x^5 + 24678266*x^6 +...
Given g.f. A(x), the table of coefficients in A(x)^n begins:
n=1: [(1), 1, 1, 8, 247, 33184, 24678266, 125237615376, ...];
n=2: [1,(2), 3, 18, 511, 66878, 49423458, 250524657604, ...];
n=3: [1, 3, (6), 31, 795, 101109, 74236366, 375861227934, ...];
n=4: [1, 4, 10,(48), 1103, 135908, 99117818, 501247428704, ...];
n=5: [1, 5, 15, 70,(1440), 171311, 124068685, 626683363390, ...];
n=6: [1, 6, 21, 98, 1812,(207360), 149089887, 752169136662, ...];
n=7: [1, 7, 28, 133, 2226, 244104,(174182400), 877704854447, ...];
n=8: [1, 8, 36, 176, 2690, 281600, 199347264,(1003290624000), ...]; ...
in which the main diagonal generates the superfactorials (A000178):
[1/1, 2/2, 6/3, 48/4, 1440/5, 207360/6, 174182400/7, 1003290624000/8, ...].
The logarithmic derivative of the superfactorials forms another diagonal:
A219266 = [1, 3, 31, 1103, 171311, 149089887, 877704854447, ...].

Crossrefs

Cf. A219266, A000178, A219270.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*sum(k=0,n+1,prod(j=0,k,j!)*x^k)+x^2*O(x^n)),n)}
for(n=0,21,print1(a(n),", "))

Formula

G.f. A(x) satisfies: [x^n] A(x)^(n+1)/(n+1) = Product_{k=0..n} k! = superfactorial A000178(n).
G.f.: x / Series_Reversion(x*F(x)) where F(x) = Sum_{n>=0} x^n*Product_{k=0..n} k! is the g.f. of A000178.

A219267 Logarithmic derivative of the hyperfactorials (A002109).

Original entry on oeis.org

1, 7, 313, 110143, 431860201, 24185951471887, 23238336572015738041, 445571476975584446962639039, 194201470505208674769594891331807753, 2157794122078406207016487628429579826176795887, 677208230450612019931822374477208301572175793625037599321
Offset: 1

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Hyperfactorial A002109(n) = Product_{k=0..n} k^k.

Examples

			L.g.f.: L(x) = x + 7*x^2/2 + 313*x^3/3 + 110143*x^4/4 + 431860201*x^5/5 +...
where
exp(L(x)) = 1 + x + 4*x^2 + 108*x^3 + 27648*x^4 + 86400000*x^5 + 4031078400000*x^6 +...+ n^n*(n-1)^(n-1)*(n-2)^(n-2)*...*3^3*2^2*1^1*0^0**x^n +...

References

  • Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Crossrefs

Cf. A002109, A219266, A219268.

Programs

  • Mathematica
    nmax=15; Rest[CoefficientList[Series[Log[Sum[Product[j^j,{j,1,k}]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)
  • PARI
    {a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ A * n^(n*(n+1)/2 + 13/12) / exp(n^2/4), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
Showing 1-3 of 3 results.

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