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.

A317344 O.g.f. A(x) satisfies: [x^n] exp( n^2*x - n*A(x) ) = 0 for n >= 1.

Original entry on oeis.org

1, 1, 6, 74, 1400, 35676, 1140328, 43740848, 1954336608, 99561874080, 5691393582336, 360561583177440, 25064797000415744, 1896477768217789120, 155128714525468598400, 13639839907494680219648, 1282811359778733608826368, 128498290985443181787800064, 13657938489514600713859515392, 1535272989503239280608301470720, 181975961346350933380240113192960
Offset: 1

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 74*x^4 + 1400*x^5 + 35676*x^6 + 1140328*x^7 + 43740848*x^8 + 1954336608*x^9 + 99561874080*x^10 + ...
such that [x^n] exp( n^2*x - n*A(x) ) = 0  for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*x - n*A(x) ) begins:
n=1: [1, 0, -2, -36, -1764, -167280, -25620600, -5737974480, ...];
n=2: [1, 2, 0, -88, -4160, -371328, -55329536, -12201990400, ...];
n=3: [1, 6, 30, 0, -7812, -698184, -97733304, -20791334880, ...];
n=4: [1, 12, 136, 1296, 0, -1171968, -168658176, -33909447168, ...];
n=5: [1, 20, 390, 7220, 113020, 0, -265712600, -55963975600, ...];
n=6: [1, 30, 888, 25704, 709056, 16600320, 0, -84622337280, ...];
n=7: [1, 42, 1750, 72072, 2909340, 112245672, 3684715944, 0, ...];
n=8: [1, 56, 3120, 172640, 9455488, 508540416, 26199517696, 1150524892160, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1945*x^4/4! + 178041*x^5/5! + 26792971*x^6/6! + 5940440563*x^7/7! + ... + A317343(n)*x^n/n! + ...

Links

Crossrefs

Cf. A317343, A317346, A317347.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = (2 + LambertW(-2*exp(-2))) * sqrt(-LambertW(-2*exp(-2)) * (1 + LambertW(-2*exp(-2)))) / (4*sqrt(2)*Pi) = 0.0440433939... - Vaclav Kotesovec, Aug 06 2018

A317345 E.g.f. A(x) satisfies: [x^n] exp(n^3*x) / A(x)^(n^2) = 0 for n >= 1.

Original entry on oeis.org

1, 1, 5, 445, 196105, 221673401, 501981700621, 1983064113021685, 12488526496641458705, 117611695946767352571505, 1578802193598207376026165781, 29098684071572000208903027320621, 714476480265312671332820625804579865, 22796869288656035590303941174243615386665, 925701505348044648968634173494720540556875805
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2018

Keywords

Comments

It is remarkable that the logarithm of the e.g.f. A(x) should be an integer series.
Periodic modulo 10: a(5*n+k) = [1,1,5,5,5](k) (mod 10), for n>=0 and k = 0..4 (conjecture).

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + 12488526496641458705*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x)^(n^2) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ... + A317346(n)*x^n + ...

Links

Crossrefs

Cf. A317343, A317346.

Programs

  • PARI
    {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ sqrt(1-c) * 3^(3*n - 7/3) * n^(3*n - 2) / (exp(3*n) * c^(n - 1/3) * (3-c)^(2*n - 2)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 07 2018

A319144 E.g.f. A(x) satisfies: [x^(n-1)] exp(n^2*x) / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 2, 7, 82, 3413, 310306, 47180827, 10609392242, 3284088709897, 1333647722701378, 686179134994911311, 435599748089861536402, 334122749226062422725277, 304457064400271021354494562, 324970210527067394401358110243, 401523372735670670696974799321266, 568490192646838149936392483264664977, 914248292513326978923735948784457567362
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2018

Keywords

Comments

It is remarkable that the logarithmic derivative of the e.g.f. A(x) should be an integer series.

Examples

			E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 82*x^3/3! + 3413*x^4/4! + 310306*x^5/5! + 47180827*x^6 + 10609392242*x^7/7! + 3284088709897*x^8/8! + 1333647722701378*x^9/9! + 686179134994911311*x^10/10! + 435599748089861536402*x^11/11! + 334122749226062422725277*x^12/12! +...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n begins
n=1: [1, -1, -2, -48, -2616, -262080, -41718240, -9630270720, ...];
n=2: [1, 0, -6, -112, -5592, -547968, -86345120, -19809990912, ...];
n=3: [1, 3, 0, -222, -10728, -958824, -144971712, -32519314080, ...];
n=4: [1, 8, 52, 0, -18648, -1693248, -236690784, -50727983616, ...];
n=5: [1, 15, 210, 2420, 0, -2739720, -399251600, -80125144800, ...];
n=6: [1, 24, 558, 12192, 221184, 0, -616918320, -131299591680, ...];
n=7: [1, 35, 1204, 40278, 1272768, 33597312, 0, -196436730672, ...];
n=8: [1, 48, 2280, 106688, 4869552, 210771456, 7654459648, 0, ...]; ...
in which the n-th term in row n forms a diagonal of zeros after an initial '1'.
RELATED SERIES.
The logarithmic derivative of the e.g.f. appears to be an integer series:
A'(x)/A(x) = 2 + 3*x + 28*x^2 + 475*x^3 + 11556*x^4 + 362418*x^5 + 13820696*x^6 + 617990499*x^7 + 31613351140*x^8 + 1817581003238*x^9 + ... + A319146(n+1)*x^n + ...

Links

Crossrefs

Cf. A319146, A317343, A303060.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( exp(m^2*x +x*O(x^n))/Ser(A)^(m) )[m]/m ); H=A;A[n+1]}
for(n=0,21, print1(a(n)*n!,", "))

Formula

a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(2*n - 1) / (exp(2*n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Aug 11 2021
Showing 1-3 of 3 results.

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