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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A296170 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2) for n>=1.

Original entry on oeis.org

1, 1, -1, -11, -239, -17059, -2145689, -412595231, -111962826751, -40590007936199, -18900753214178609, -10974885891916507219, -7765167486697279401071, -6571694718107813687003051, -6551841491106355785902247049, -7597507878436131044487467850599, -10136619271768255373949409579309439, -15416099624633773180711565727641136271, -26508391106594400233543066679525341764961
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

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

Examples

			E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! - 2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! - 40590007936199*x^9/9! - 18900753214178609*x^10/10! - 10974885891916507219*x^11/11! - 7765167486697279401071*x^12/12! - 6571694718107813687003051*x^13/13! - 6551841491106355785902247049*x^14/14! - 7597507878436131044487467850599*x^15/15! +...
To illustrate [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2), form a table of coefficients of x^k in A(x)^(n^2) that begins as
n=1: [(1), (1), -1/2, -11/6, -239/24, -17059/120, -2145689/720, ...];
n=2: [1, (4), (4), -28/3, -196/3, -10472/15, -614264/45, ...];
n=3: [1, 9, (63/2), (63/2), -1701/8, -98217/40, -3168081/80, ...];
n=4: [1, 16, 112, (1232/3), (1232/3), -95648/15, -4835264/45, ...];
n=5: [1, 25, 575/2, 11725/6, (190225/24), (190225/24), ...];
n=6: [1, 36, 612, 6444, 45684, (1043784/5), (1043784/5), ...];
n=7: [1, 49, 2303/2, 102949/6, 4313617/24, 164086349/120, (5086480231/720), (5086480231/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^2) by n^2, we obtain the following sequence:
[1, 1, 7, 154, 7609, 695856, 103805719, 23134327168, 7227250033329, 3017857024161280, 1623903877812828871, ..., A296232(n), ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 - 478780244956262592748*x^16 - 42865943103053965559668*x^17 - 4097785410628237071311764*x^18 - 416572537937169684523985420*x^19 - 44873737158384968851319470220*x^20 +...+ A296171(n)*x^n +...

Links

Crossrefs

Cf. A296171 (log), A296232, A296172, A296174, A296176, A182962.

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] = (V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

The logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = Sum{n>=1} A296171(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^2) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^2) for n>=1, when evaluated at x = 0.
a(n) ~ c * d^n * n^(2*n-2) / exp(2*n), where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = -0.1875440087... - Vaclav Kotesovec, Dec 23 2017

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

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

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

A296173 G.f. equals the logarithm of the e.g.f. of A296172.

Original entry on oeis.org

1, -3, -30, -2686, -517311, -173118807, -88535206152, -63977172334344, -61971659588102940, -77470793599569049440, -121439997599825393413344, -233353875172602479932391040, -539638027429765922735002220880, -1479049138515818646669055218090480, -4742815067612592169849894663392228480, -17597031102801426396121130730318359114880, -74817150772352720408567833273371047298417408
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.

Examples

			G.f. A(x) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 - 539638027429765922735002220880*x^13 - 1479049138515818646669055218090480*x^14 - 4742815067612592169849894663392228480*x^15 +...
such that
G(x) = exp(A(x)) = 1 + x - 5*x^2/2! - 197*x^3/3! - 65111*x^4/4! - 62390159*x^5/5! - 125012786669*x^6/6! - 447082993406405*x^7/7! - 2583111044504384687*x^8/8! - 22511408975342644804991*x^9/9! - 281350305428215911326408789*x^10/10! - 4850582201056517165575319399909*x^11/11! - 111834955668396093904661955538037255*x^12/12! +...
satisfies [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.
Series_Reversion(A(x)) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...+ A295813(n)*x^n +...

Links

Crossrefs

Cf. A296172, A295813, A296171, A296175, A296177.

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] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ -sqrt(1-c) * 3^(3*n - 3) * n^(2*n - 7/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 3) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A296175 G.f. equals the logarithm of the e.g.f. of A296174.

Original entry on oeis.org

1, -7, -493, -341101, -680813601, -2923660883625, -22996362478599551, -299331006952284448127, -6006951481145880962408552, -176288642409787912257773903552, -7260231964238768891891716773249396, -405879958110794676900559524931590299892, -29968312587171485511980894312242331299164248, -2855987647850204274493781603297327940940773633392
Offset: 1

Views

Author

Paul D. Hanna, Dec 07 2017

Keywords

Comments

E.g.f. G(x) of A296174 satisfies: [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.

Examples

			G.f. A(x) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 - 176288642409787912257773903552*x^10 - 7260231964238768891891716773249396*x^11 - 405879958110794676900559524931590299892*x^12 +...
such that
G(x) = exp(A(x)) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! - 639738016495616440994202167765715629*x^10/10! +...
satisfies [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
Series_Reversion(A(x)) = x + 7*x^2 + 591*x^3 + 360071*x^4 + 696409901*x^5 + 2958728428011*x^6 + 23164541753169117*x^7 + 300801581861406441263*x^8 +...

Links

Crossrefs

Cf. A296174, A296171, A296173, A296177.

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] = (V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) ~ -sqrt(1-c) * 2^(8*n - 17/2) * n^(3*n - 9/2) / (sqrt(Pi) * c^n * (4-c)^(3*n - 4) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.07930960512711365643910864... - Vaclav Kotesovec, Oct 13 2020

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.
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