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.

A303061 G.f. A(x) satisfies: [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 1, 1, 7, 87, 1667, 42971, 1387941, 53739797, 2421203261, 124265293581, 7150727869627, 455701200668539, 31846907669892495, 2421141672213472919, 198897819736367366729, 17556316040185549675881, 1656973308228250148662329, 166509657562826568857464281, 17749793745710561363581851663, 2000554650909636157531234301439
Offset: 0

Views

Author

Paul D. Hanna, Apr 17 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 87*x^4 + 1667*x^5 + 42971*x^6 + 1387941*x^7 + 53739797*x^8 + 2421203261*x^9 + 124265293581*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1+x)^(n*(n-1)) / A(x)^n begins:
n=1: [1, -1, 0, -6, -74, -1500, -39688, -1302742, ...];
n=2: [1, 0, -2, -12, -159, -3136, -82180, -2680752, ...];
n=3: [1, 3, 0, -26, -300, -5454, -137764, -4398210, ...];
n=4: [1, 8, 24, 0, -548, -9576, -223760, -6847536, ...];
n=5: [1, 15, 100, 350, 0, -16022, -376660, -10771830, ...];
n=6: [1, 24, 270, 1844, 7641, 0, -596908, -17643792, ...];
n=7: [1, 35, 588, 6258, 46186, 224196, 0, -26940146, ...];
n=8: [1, 48, 1120, 16864, 182640, 1478160, 8281968, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1.

Links

Crossrefs

Cf. A303060.

Programs

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

Formula

a(n) + a(n-1) = A303060(n) for n>=0.
a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.06049920104... - Vaclav Kotesovec, Oct 06 2020

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

A319143 G.f. A(x) satisfies: [x^(n-1)] (1+x)^(n^3) / A(x)^(n^2) = 0 for n>1.

Original entry on oeis.org

1, 2, 5, 190, 24444, 6189050, 2551526428, 1545212826174, 1288051774444110, 1412705106844118046, 1971892031185697252554, 3413903325218336008192250, 7181500581229611492081984526, 18048175081484797766245697300090, 53425898749319275351535742806432314, 184046820557885265127311961578368691278, 730192327635057505047728578528016106455194
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 190*x^3 + 24444*x^4 + 6189050*x^5 + 2551526428*x^6 + 1545212826174*x^7 + 1288051774444110*x^8 + ...
The table of coefficients of x^k in (1+x)^(n^3) / A(x)^(n^2) begins:
n=1: [1, -1, -3, -179, -23881, -6115379, -2532879873, ...];
n=2: [1, 0, -16, -728, -96144, -24548304, -10154627640, ...];
n=3: [1, 9, 0, -1878, -231876, -57206466, -23347565964, ...];
n=4: [1, 32, 432, 0, -472008, -111871136, -43940424080, ...];
n=5: [1, 75, 2675, 55475, 0, -199916560, -76768966500, ...];
n=6: [1, 144, 10152, 460056, 13896684, 0, -126293662512, ...];
n=7: [1, 245, 29694, 2364152, 137272471, 5735706025, 0, ...]; ...
in which the n-th term in row n forms a diagonal of zeros after an initial '1'.

Crossrefs

Cf. A303060.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( (1+x +x*O(x^n))^(m^3)/Ser(A)^(m^2) )[m]/m^2 ); A[n+1]}
for(n=0,30, print1(a(n),", "))
Showing 1-3 of 3 results.

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