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 1-4 of 4 results.

A300597 O.g.f. A(x) satisfies: [x^n] exp( n^4 * A(x) ) = n^4 * [x^(n-1)] exp( n^4 * A(x) ) for n>=1.

Original entry on oeis.org

1, 8, 2187, 2351104, 6153518125, 31779658925496, 287364845865893467, 4200677982722915635200, 93566442152660422280250537, 3030525904161802498705606745000, 137355046868929476532154243693393581, 8436685562091750543736612601781557411328, 683522945769518614776208838188411394718328617
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare to: [x^n] exp( n^4 * x ) = n^3 * [x^(n-1)] exp( n^4 * x ) for n>=1.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + 31779658925496*x^6 + 287364845865893467*x^7 + 4200677982722915635200*x^8 + ...
where
exp(A(x)) = 1 + x + 17*x^2/2! + 13171*x^3/3! + 56479849*x^4/4! + 738706542221*x^5/5! + 22885801082965201*x^6/6! + 1448479282286023114807*x^7/7! + ... + A300596(n)*x^n/n! + ...
such that: [x^n] exp( n^4 * A(x) ) = n^4 * [x^(n-1)] exp( n^4 * A(x) ).

Links

Crossrefs

Cf. A300596, A296171, A300591, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = ((#A-1)^4*V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300596.

A300614 E.g.f. A(x) satisfies: [x^n] A(x)^(n^5) = n^5 * [x^(n-1)] A(x)^(n^5) for n>=1.

Original entry on oeis.org

1, 1, 33, 118195, 3419881993, 412433022394701, 152749066271797582081, 143430189975946314906194983, 297572051428536567500380512047505, 1228369468294423956894049108209998483353, 9295358239339907973775754707697954813272247041, 120806095217585335844962641542342569940874366294995451
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^(n^5) = n^4 * [x^(n-1)] exp(x)^(n^5) for n>=1.

Examples

			E.g.f.: A(x) = 1 + x + 33*x^2/2! + 118195*x^3/3! + 3419881993*x^4/4! + 412433022394701*x^5/5! + 152749066271797582081*x^6/6! + 143430189975946314906194983*x^7/7! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^(n^5) begins:
n=1: [(1), (1), 33/2, 118195/6, 3419881993/24, 137477674131567/40, ...];
n=2: [1, (32), (1024), 1955104/3, 13739402240/3, 1651861749195104/15, ...];
n=3: [1, 243, (66825/2), (16238475/2), 288411062643/8, 33749327928610701/40, ...];
n=4: [1, 1024, 540672, (647668736/3), (663212785664/3), 18460138990560256/5, ...];
n=5: [1, 3125, 9865625/2, 31824134375/6, (116555654565625/24), (364236420517578125/24), ...];
n=6: [1, 7776, 30357504, 79484677920, 158407197944832, (1433574291388125024/5), (11147473689834060186624/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 1024 = 2^5*32; 16238475/2 = 3^5*66825/2; 663212785664/3 = 4^5*647668736/3; ...
illustrating that: [x^n] A(x)^(n^5) = n^5 * [x^(n-1)] A(x)^(n^5).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 16*x^2 + 19683*x^3 + 142475264*x^4 + 3436799053125*x^5 + 212148041589128016*x^6 + 28458158819417861315152*x^7 + 7380230750280159370894934016*x^8 + ... + A300615(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300615.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = ((#A-1)^5*V[#A-1] - V[#A])/(#A-1)^5 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300615(n)*x^n, a power series in x with integer coefficients.

A300616 E.g.f. A(x) satisfies: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n for n>=1.

Original entry on oeis.org

1, 1, 7, 199, 14065, 1924201, 445859911, 161145717727, 85790577700129, 64427620614173425, 65943035132156264071, 89425725156530626400791, 156922032757769223085752337, 349233620942232034199096926489, 968890106809715834110637461124935, 3301188169350221687517822373590448111, 13634136452997022097853039839798901714241
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.
Compare to e.g.f. G(x) of A182962: [x^n] G(x)^n = n * [x^(n-1)] G(x)^n for n>=1.

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14065*x^4/4! + 1924201*x^5/5! + 445859911*x^6/6! + 161145717727*x^7/7! + 85790577700129*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^n begins:
n=1: [(1), (1), 7/2, 199/6, 14065/24, 1924201/120, 445859911/720, ...];
n=2: [1, (2), (8), 220/3, 3752/3, 502114/15, 57409744/45, ...];
n=3: [1, 3, (27/2), (243/2), 16035/8, 2098161/40, 157765131/80, ...];
n=4: [1, 4, 20, (536/3), (8576/3), 1096868/15, 121987336/45, ...];
n=5: [1, 5, 55/2, 1475/6, (91825/24), (2295625/24), 503279435/144, ...];
n=6: [1, 6, 36, 324, 4920, (601074/5), (21638664/5), 7491519768/35...];
n=7: [1, 7, 91/2, 2485/6, 147721/24, 17641687/120, (3752979139/720), (183895977811/720), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 8 = 2^2*2; 243/2 = 3^2*27/2; 8576/3 = 4^2*536/3; ...
illustrating that: [x^n] A(x)^n = n^2 * [x^(n-1)] A(x)^n.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + 2093655600*x^8 + 175312873125*x^9 + 17987972309725*x^10 + ... + A300617(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300594, A300596, A300614, A300617.

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)^2*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} A300617(n)*x^n, a power series in x with integer coefficients.
a(n) ~ c * (n!)^3, where c = 1.685041722777551007711429045295022018562828... - Vaclav Kotesovec, Mar 10 2018

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.
Showing 1-4 of 4 results.

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