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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A300873 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

Original entry on oeis.org

1, 1, 3, 43, 2041, 197721, 31094251, 7086479443, 2187876597873, 874871971357681, 438740658523346131, 269314248304239932091, 198529013874402868930153, 173067121551267519897494473, 176154202119865662835343738811, 207099741506845262022248534098531, 278645958801870115911315221474653921, 425605862347493892454320041743878801633
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

Compare to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
illustrating that: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ... + A300874(n)*x^n + ...

Links

Crossrefs

Cf. A300874, A300870, A295811, A300590, A296170, A182962.

Programs

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

Formula

a(n) ~ c * d^n * n!^2 / n^3, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.75891265... - Vaclav Kotesovec, Aug 11 2021

A300618 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n for n>=1.

Original entry on oeis.org

1, 1, 15, 1285, 347065, 224232501, 296201195791, 719274160258585, 2967337954539761265, 19563048191912257746505, 196302561889372679184550831, 2881342883089548932078551914861, 59862434550069057805236434063104105, 1712289828911477479390772271103153886845
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^n = [x^(n-1)] exp(x)^n for n>=1.

Examples

			E.g.f.: A(x) = 1 + x + 15*x^2/2! + 1285*x^3/3! + 347065*x^4/4! + 224232501*x^5/5! + 296201195791*x^6/6! + 719274160258585*x^7/7! + 2967337954539761265*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 15/2, 1285/6, 347065/24, 74744167/40, ...];
n=2: [1, (2), (16), 1330/3, 88220/3, 56540144/15, ...];
n=3: [1, 3, (51/2), (1377/2), 358875/8, 228121101/40, ...];
n=4: [1, 4, 36, (2852/3), (182528/3), 38352496/5, ...];
n=5: [1, 5, 95/2, 7385/6, (1857145/24), (232143125/24), ...];
n=6: [1, 6, 60, 1530, 94500, (58551624/5), (12647150784/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 16 = 2^3*2; 1377/2 = 3^3*51/2; 182528/3 = 4^3*2852/3; ...
illustrating that: [x^n] A(x)^n = n^3 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + ... + A300619(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300614, A300616, A300619.

Programs

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

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300619(n)*x^n, a power series in x with integer coefficients.
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