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 1-10 of 13 results. Next

A300598 a(n) = A300591(n) / n for n>=1.

Original entry on oeis.org

1, 1, 9, 184, 6105, 285909, 17599855, 1366487208, 130312110537, 14977420657205, 2044343858148526, 327321396575462328, 60816978336055883851, 12988287586752637095951, 3161098207809674432144760, 870056671853543460441640960, 268957112246197632099231284121, 92799365448465489168470692401021, 35538203127994691420731763316052499, 15028828305943284874962308136851532840
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

If G(x) satisfies: [x^n] exp (n^2 * G(x) ) = n^2 * [x^(n-1)] exp( n^2 * G(x) ) for n>=1, then G(x) equals the o.g.f. of A300591.

Links

Crossrefs

Cf. A300591.

Programs

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

A300590 E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2) for n>=1.

Original entry on oeis.org

1, 1, 5, 175, 18385, 3759701, 1258735981, 630063839035, 445962163492385, 429694421369414185, 547875295770399220981, 903754519692129905068391, 1892423689107542226463430065, 4948056864672913520114055888445, 15922007799835205487157437619131485, 62245856465769048392433555378169339891, 292266373167286246870149657443033728860481
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^(n^2) = n * [x^(n-1)] exp(x)^(n^2) for n>=1.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + 429694421369414185*x^9/9! + 547875295770399220981*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [(1), (1), 5/2, 175/6, 18385/24, 3759701/120, 1258735981/720, ...];
n=2: [1, (4), (16), 452/3, 10448/3, 2037388/15, 333368656/45, ...];
n=3: [1, 9, (117/2), (1053/2), 79803/8, 14107743/40, 1472857749/80, ...];
n=4: [1, 16, 160, (4880/3), (78080/3), 11770672/15, 1707161056/45, ...];
n=5: [1, 25, 725/2, 27175/6, (1642225/24), (41055625/24), ...];
n=6: [1, 36, 720, 11340, 180720, (19548324/5), (703739664/5),  ...];
n=7: [1, 49, 2597/2, 154399/6, 11125009/24, (1138996229/120), (205943018701/720), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*(1); 16 = 2^2*(4); 1053/2 = 3^2*(117/2); 78080/3 = 4^2*(4880/3); 41055625/24 = 5^2*(1642225/24); ...
illustrating that: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ... + A300591(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A300870, A296170, A300591, A300592, A300594.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300591(n)*x^n, a power series in x with integer coefficients.
a(n) ~ c * n!^3 * n^2, where c = 0.1354708370957778563796... - Vaclav Kotesovec, Oct 13 2020

A300595 O.g.f. A(x) satisfies: [x^n] exp( n^3 * A(x) ) = n^3 * [x^(n-1)] exp( n^3 * A(x) ) for n>=1.

Original entry on oeis.org

1, 4, 243, 40448, 12519125, 6111917748, 4308276119854, 4151360558858752, 5268077625693186225, 8567999843251994553500, 17491034862909191177473132, 44081002571786307016424642880, 135294991782833277819666508563622, 499980220350805674732349875409752344, 2203045016526580123761644939382016407000, 11476028442989415865296132639050660100915200
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare to: [x^n] exp( n^3 * x ) = n^2 * [x^(n-1)] exp( n^3 * x ) for n>=1.
It is conjectured that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + 4308276119854*x^7 + 4151360558858752*x^8 + 5268077625693186225*x^9 + ...
where
exp(A(x)) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 976825*x^4/4! + 1507281021*x^5/5! + 4409747597401*x^6/6! + 21744850191313999*x^7/7! + ... + A300594(n)*x^n/n! + ...
such that: [x^n] exp( n^3 * A(x) ) = n^3 * [x^(n-1)] exp( n^3 * A(x) ).

Links

Crossrefs

Cf. A300594, A296171, A300591, A300593.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300594.
a(n) ~ c * n!^3 * n^3, where c = 0.40774346023... - Vaclav Kotesovec, Oct 14 2020

A300593 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^3 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

Original entry on oeis.org

1, 6, 216, 18016, 2718575, 667151244, 249904389518, 136335045655680, 104258627494173747, 108236370325030253850, 148475074256982964816314, 263023328027145941803648512, 590040725672004981627313856146, 1648073412972421008768279297745708, 5648002661974709728272920853918580200, 23444503972399728196572891896057248430080
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare to: [x^n] exp( n^2 * x ) = n * [x^(n-1)] exp( n^2 * x ) for n>=1.
It is conjectured that this sequence consists entirely of integers.

Examples

			O.g.f.: A(x) = x + 6*x^2 + 216*x^3 + 18016*x^4 + 2718575*x^5 + 667151244*x^6 + 249904389518*x^7 + 136335045655680*x^8 + 104258627494173747*x^9 ...
where
exp(A(x)) = 1 + x + 13*x^2/2! + 1333*x^3/3! + 438073*x^4/4! + 328561681*x^5/5! + 482408372341*x^6/6! + 1262989939509733*x^7/7! + ... + A300592(n)*x^n/n! + ...
such that: [x^n] exp( n^2 * A(x) ) = n^3 * [x^(n-1)] exp( n^2 * A(x) ).

Links

Crossrefs

Cf. A300592, A296171, A300591, A300595.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300592.
a(n) ~ c * n!^3, where c = 3.10566781078993955626127892108166... - Vaclav Kotesovec, Oct 14 2020

A300736 O.g.f. A(x) satisfies: A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).

Original entry on oeis.org

1, 1, 4, 24, 184, 1672, 17296, 198800, 2499200, 33992000, 496281344, 7731823616, 127946465280, 2240485196800, 41387447564800, 804353715776000, 16408115358117888, 350584123058300928, 7831051680901885952, 182550106828365115392, 4433782438058087202816, 112031844502468602085376, 2940834866411162315849728
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300735.
The e.g.f. G(x) of A300735 satisfies: [x^n] G(x)^(2*n) = (n+1) * [x^(n-1)] G(x)^(2*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 184*x^5 + 1672*x^6 + 17296*x^7 + 198800*x^8 + 2499200*x^9 + 33992000*x^10 + 496281344*x^11 + 7731823616*x^12 + ...
where
A(x) = x*(1 - x*A'(x)) / (1 - 2*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 697*x^4/4! + 25761*x^5/5! + 1371691*x^6/6! + 97677343*x^7/7! + 8869533681*x^8/8! + 993709302337*x^9/9! + 134086553693011*x^10/10! + ... + A300735(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 12*x^2 + 96*x^3 + 920*x^4 + 10032*x^5 + 121072*x^6 + 1590400*x^7 + 22492800*x^8 + 339920000*x^9 + 5459094784*x^10 + ...

Links

Crossrefs

Cf. A088716, A300987, A300989, A300991, A300993.
Cf. A300735, A088716, A300591, A296171, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x*(1-x*A')/(1-2*x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(2*(#A-1))); A[#A] = ((#A)*V[#A-1] - V[#A])/(2*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 2*n * A(x) ) = (n + 1) * [x^(n-1)] exp( 2*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^3, where c = 0.0087891365985... - Vaclav Kotesovec, Mar 20 2018

A300989 O.g.f. A(x) satisfies: A(x) = x*(1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).

Original entry on oeis.org

1, 1, 6, 50, 520, 6312, 86080, 1288704, 20862720, 361454720, 6652338176, 129341001216, 2645494627328, 56734280221696, 1272300911597568, 29769957834147840, 725430667245355008, 18379623419316338688, 483476314203202945024, 13187069277429966733312, 372512001057014648537088, 10886129458069912361631744, 328776894530826384975593472
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300988.
The e.g.f. G(x) of A300988 satisfies: [x^n] G(x)^(4*n) = (n+3) * [x^(n-1)] G(x)^(4*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ...
where
A(x) = x * (1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ... + A300988(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 18*x^2 + 200*x^3 + 2600*x^4 + 37872*x^5 + 602560*x^6 + 10309632*x^7 + 187764480*x^8 + 3614547200*x^9 + ...

Links

Crossrefs

Cf. A300988, A088716, A300736, A300987, A300991, A300993.
Cf. A300591, A296171, A300593, A300595.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = x*(1-3*x*A')/(1-4*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    /* [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^7, where c = 0.00000132855349... - Vaclav Kotesovec, Mar 20 2018

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.

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

A300871 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ) for n>=1.

Original entry on oeis.org

1, 3, 48, 1510, 71280, 4511808, 361640832, 35516910960, 4184770003200, 582762638275840, 94800017774905344, 17836975939663156224, 3847898790157443653632, 944223655310222217584640, 261663903298936561335828480, 81353978185283974468642093056, 28208743160867030634605718994944, 10849126423364041648181194666082304, 4605289001051501407092469612444385280
Offset: 1

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

Compare to: [x^n] exp( n*(n+1) * x ) = (n+1) * [x^(n-1)] exp( n*(n+1) * x ) for n>=1.
O.g.f. equals the logarithm of the e.g.f. of A300870.
The e.g.f. G(x) of A300870 satisfies: [x^n] G(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] G(x)^(n*(n+1)) for n>=1.
It is conjectured that this sequence consists entirely of integers.
a(n) is divisible by n*(n+1)/2 (conjecture); a(n) = n*(n+1)/2 * A300872(n).

Examples

			O.g.f.: A(x) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ...
where
exp(A(x)) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + ... + A300870(n)*x^n/n! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp( n*(n+1) * A(x) ) begins:
n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];
n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];
n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];
n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];
n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];
n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];
n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...
illustrating: [x^n] exp( n*(n+1) * A(x) ) = n*(n+1) * [x^(n-1)] exp( n*(n+1) * A(x) ).
The values A300872(n) = a(n) / (n*(n+1)/2) begin:
[1, 1, 8, 151, 4752, 214848, 12915744, 986580860, 92994888960, ...]
and appear to consist entirely of integers.

Links

Crossrefs

Cf. A300870, A300872, A300591, A296171, A300874.

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] = ((#A-1)*(#A)*V[#A-1] - V[#A])/(#A-1)/(#A) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

A300619 O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ) for n>=1.

Original entry on oeis.org

1, 7, 207, 14226, 1852800, 409408077, 142286748933, 73448832515952, 53835885818473473, 54041298732304775000, 72129250579997923194091, 124900802377559946754633602, 274851919918333747166200590840, 755158633069275870471471631726803, 2551279948230221759814139760682442500
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300618.
It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + 142286748933*x^7 + 73448832515952*x^8 + 53835885818473473*x^9 + ...
where
exp(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! + ... + A300618(n)*x^n/n! + ...
such that: [x^n] exp( n * A(x) ) = n^3 * [x^(n-1)] exp( n * A(x) ).

Links

Crossrefs

Cf. A300618, A296171, A300591, A300593, A300595, A300597, 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)^3*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))
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