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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A300591 O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ) for n>=1.

Original entry on oeis.org

1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, 1172808994833, 149774206572050, 22487782439633786, 3927856758905547936, 790620718368726490063, 181836026214536919343314, 47416473117145116482171400, 13920906749656695367066255360, 4572270908185359745686931830057, 1670388578072378805032472463218378, 675225859431899136993903503004997481, 300576566118865697499246162737030656800
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.
a(n) is divisible by n (conjecture): A300598(n) = a(n)/n for n>=1.

Examples

			O.g.f.: 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 + ...
where
exp(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! + ... + A300590(n)*x^n/n! + ...
such that: [x^n] exp( n^2 * A(x) ) = n^2 * [x^(n-1)] exp( n^2 * A(x) ).

Links

Crossrefs

Cf. A300590, A300598, A300871, A296171, A300593, 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)^2*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 A300590.
a(n) ~ c * n!^2 * n^2, where c = 0.1354708370957778563796... - Vaclav Kotesovec, Oct 13 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

A300594 E.g.f. A(x) satisfies: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3) for n>=1.

Original entry on oeis.org

1, 1, 9, 1483, 976825, 1507281021, 4409747597401, 21744850191313999, 167557834535988306033, 1913194223179191462419065, 31110747474489521617502800201, 698529144858380953105954686101811, 21123268203104470199318422678044241129, 842759726425517953579189712209822358428213, 43599233739340643789919321494623289001407934105
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^(n^3) = n^2 * [x^(n-1)] exp(x)^(n^3) for n>=1.

Examples

			E.g.f.: 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! + 167557834535988306033*x^8/8! + 1913194223179191462419065*x^9/9! + 31110747474489521617502800201*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^3) begins:
  n=1: [(1), (1), 9/2, 1483/6, 976825/24, 502427007/40, 4409747597401/720, ...]
  n=2: [1, (8), (64), 6856/3, 1022528/3, 1543097816/15, 2237393526784/45, ...]
  n=3: [1, 27, (945/2), (25515/2), 10692675/8, 14849374869/40, 13978534445001/80, ...]
  n=4: [1, 64, 2304, (226880/3), (14520320/3), 5124803136/5, 20241220116736/45, ...]
  n=5: [1, 125, 16625/2, 2510375/6, (553359625/24), (69169953125/24), ...];
  n=6: [1, 216, 24192, 1918728, 131302080, (56555402904/5), (12215967027264/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 64 = 2^3*8; 25515/2 = 3^3*945/2; 14520320/3 = 4^3*226880/3; ...
illustrating that: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(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 + 8567999843251994553500*x^10 + ... + A300595(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300595.

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 ); EGF=Ser(A); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300595(n)*x^n, a power series in x with integer coefficients.
a(n) ~ c * n!^4 * n^3, where c = 0.40774346023... - 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

A300987 O.g.f. A(x) satisfies: A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).

Original entry on oeis.org

1, 1, 5, 36, 327, 3489, 42048, 559008, 8073243, 125328411, 2075525505, 36460943208, 676484058564, 13210384019292, 270753854165604, 5810388957096552, 130292809125319539, 3047472204302259711, 74227110587569392471, 1879966895740420683492, 49443968787368161215087, 1348661750106914651234385, 38107004920979745293594856, 1114125483618428275543280400, 33669232396216806674333898900
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300986.
The e.g.f. G(x) of A300986 satisfies: [x^n] G(x)^(3*n) = (n+2) * [x^(n-1)] G(x)^(3*n) for n>=1.

Examples

			O.g.f.: A(x) = x + x^2 + 5*x^3 + 36*x^4 + 327*x^5 + 3489*x^6 + 42048*x^7 + 559008*x^8 + 8073243*x^9 + 125328411*x^10 + 2075525505*x^11 + ...
where
A(x) = x*(1 - 2*x*A'(x)) / (1 - 3*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1009*x^4/4! + 44541*x^5/5! + 2799931*x^6/6! + 233188033*x^7/7! + 24562692897*x^8/8! + 3168510747769*x^9/9! + 488856473079571*x^10/10! + ... + A300986(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 15*x^2 + 144*x^3 + 1635*x^4 + 20934*x^5 + 294336*x^6 + 4472064*x^7 + 72659187*x^8 + 1253284110*x^9 + 22830780555*x^10 + ...

Links

Crossrefs

Cf. A300986, A088716, A300736, A300989, A300991, A300993.
Cf. A296171, A300593, A300595.

Programs

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

Formula

O.g.f. A(x) satisfies: [x^n] exp( 3*n * A(x) ) = (n + 2) * [x^(n-1)] exp( 3*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^5, where c = 0.00014640560804... - 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

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), ", "))

A300615 O.g.f. A(x) satisfies: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ) for n>=1.

Original entry on oeis.org

1, 16, 19683, 142475264, 3436799053125, 212148041589128016, 28458158819417861315152, 7380230750280159370894934016, 3385049575573746853297963891959753, 2561548157856026756893458765378989150000, 3026444829408778969259555715061437179090541565, 5340113530831632053993990154143996936096662034267136
Offset: 1

Views

Author

Paul D. Hanna, Mar 10 2018

Keywords

Comments

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

Examples

			O.g.f.: A(x) = x + 16*x^2 + 19683*x^3 + 142475264*x^4 + 3436799053125*x^5 + 212148041589128016*x^6 + 28458158819417861315152*x^7 + ...
where
exp(A(x)) = 1 + x + 33*x^2/2! + 118195*x^3/3! + 3419881993*x^4/4! + 412433022394701*x^5/5! + 152749066271797582081*x^6/6! + ... + A300614(n)*x^n/n! + ...
such that: [x^n] exp( n^5 * A(x) ) = n^5 * [x^(n-1)] exp( n^5 * A(x) ).

Links

Crossrefs

Cf. A300614, 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)^5)); A[#A] = ((#A-1)^5*V[#A-1] - V[#A])/(#A-1)^5 ); 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 A300614.
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