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.

A300617 O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ) for n>=1.

Original entry on oeis.org

1, 3, 30, 550, 15375, 601398, 31299268, 2093655600, 175312873125, 17987972309725, 2221603804365924, 325310016974127276, 55749742122979646105, 11056914755618659399500, 2513208049272148754203200, 649086459674801585681092992, 189044817293654530855544266209, 61671809408989968268084102641075, 22399957973327602630210233608217250, 9009223131975798265447660437783058050
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

Compare to: [x^n] exp( n * x ) = [x^(n-1)] exp( n * x ) for n>=1.
It is conjectured that this sequence consists entirely of integers.
a(n) is divisible by n*(n+1)/2 (conjecture); A300589(n) = a(n) / (n*(n+1)/2).

Examples

			O.g.f.: 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 + ...
where
exp(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! + ... + A300616(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^2 * [x^(n-1)] exp( n * A(x) ).
RELATED SEQUENCES.
The sequence A300589(n) = a(n) / (n*(n+1)/2) begins:
[1, 1, 5, 55, 1025, 28638, 1117831, 58157100, 3895841625, 327054041995, ...].
The table of coefficients in x^k/k! in exp(-n*A(x)) * (1 - n^2*x) begins:
n=1: [1, 0, 5, 178, 13269, 1853876, 434314705, 158024698350, ...];
n=2: [1, -2, 0, 248, 22976, 3416592, 822150016, 303575549440, ...];
n=3: [1, -6, -27, 0, 21861, 4129758, 1079984097, 415322613324, ...];
n=4: [1, -12, -88, -848, 0, 3286304, 1109402752, 469332346368, ...];
n=5: [1, -20, -195, -2650, -55675, 0, 794678425, 438768342850, ...];
n=6: [1, -30, -360, -5832, -161856, -6828624, 0, 293555007360, ...];
n=7: [1, -42, -595, -10892, -339339, -18549958, -1433676839, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.

Links

Crossrefs

Cf. A300616, A300589, A296171, A300591, A300593, A300595, A300597.

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

Formula

O.g.f. equals the logarithm of the e.g.f. of A300616.
O.g.f. A(x) satisfies: [x^n] exp(-n*A(x)) * (1 - n^2*x) = 0, for n > 0. - Paul D. Hanna, Oct 15 2018
a(n) ~ c * (n!)^2, where c = 1.685041722777551007711429045295022018562828... - Vaclav Kotesovec, Mar 10 2018

A300589 a(n) = A300617(n) / (n*(n+1)/2) for n>=1.

Original entry on oeis.org

1, 1, 5, 55, 1025, 28638, 1117831, 58157100, 3895841625, 327054041995, 33660663702514, 4170641243258042, 612634528823952155, 105303950053511041900, 20943400410601239618360, 4772694556432364600596272, 1235587041134996933696367753, 360653856192923791041427500825, 117894515649092645422159124253775, 42901062533218086978322192560871705
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

It is conjectured that this sequence consists entirely of integers.
O.g.f. G(x) of A300617 satisfies: [x^n] exp(n*G(x)) = n^2 * [x^(n-1)] exp(n*G(x)) for n>=1.

Links

Crossrefs

Cf. A300617, A300616.

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) ); polcoeff( log(Ser(A)), n) / (n*(n+1)/2)}
for(n=1, 20, print1(a(n), ", "))

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 1-3 of 3 results.

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