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.

Previous Showing 11-18 of 18 results.

A300988 E.g.f. A(x) satisfies: [x^n] A(x)^(4*n) = (n + 3) * [x^(n-1)] A(x)^(4*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 43, 1369, 69561, 4991371, 471516403, 56029153713, 8112993527089, 1398528216254611, 281935928284459131, 65543089930613822473, 17373185629100099938153, 5201713100466658289659419, 1745470558150260528082445251, 652016607740826946854349450081, 269558306371535265856134699842913, 122707064351998882900943162086492963, 61225312946191234549695844364141862859
Offset: 0

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

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

Examples

			E.g.f.: 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! + ...
such that [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n) for n>=1.
RELATED SERIES.
A(x)^4 = 1 + 4*x + 24*x^2/2! + 304*x^3/3! + 8320*x^4/4! + 390144*x^5/5! + 26653696*x^6/6! + 2434011136*x^7/7! + 282056564736*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(4*n) begins:
n=1: [(1), (4), 12, 152/3, 1040/3, 16256/5, 1665856/45, 152125696/315, ...];
n=2: [1, (8), (40), 592/3, 3728/3, 157376/15, 4992064/45, 86636800/63, ...];
n=3: [1, 12, (84), (504), 3264, 129408/5, 1273536/5, 104486784/35, ...];
n=4: [1, 16, 144, (3104/3), (21728/3), 283264/5, 23764096/45, 1844359168/315, ...];
n=5: [1, 20, 220, 5560/3, (42800/3), (342400/3), 9296960/9, 687731200/63, ...];
n=6: [1, 24, 312, 3024, 25680, (1073856/5), (9664704/5), 690265344/35, ...];
n=7: [1, 28, 420, 13832/3, 129248/3, 1905792/5, (156447424/45), (312894848/9), ...]; ...
in which the coefficients in parenthesis are related by
4 = 4*(1); 40 = 5*(8); 504 = 6*(84); 21728/3 = 7*(3104/3); 342400/3 = 8*(42800/3); 9664704/5 = 9*(1073856/5); ...
illustrating: [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 3*x*A'(x)/A(x)) / (1 - 4*x*A'(x)/A(x));
explicitly,
log(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 + ... + A300989(n)*x^n + ...

Links

Crossrefs

Cf. A300989, A182962, A300735, A300986, A300990, A300992.
Cf. A300735, A300870, A300590, A296170.

Programs

  • PARI
    {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)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1,n, A = exp( x*(A-3*x*A')/(A-4*x*A' +x*O(x^n)) ) ); n!*polcoeff(A,n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 3*x*A'(x)) / (A(x) - 4*x*A'(x)) ).

A300990 E.g.f. A(x) satisfies: [x^n] A(x)^(5*n) = (n+4) * [x^(n-1)] A(x)^(5*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 49, 1777, 101541, 8140411, 855134533, 112545136929, 17984228218057, 3409574126285971, 753501858876909561, 191427165598888279633, 55281557535673696196269, 17980171490246227257206667, 6535371640250591590600624141, 2637140727761043517527505819201, 1174615924949881797618432103697553, 574619225547616163988810792896019619
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 49*x^3/3! + 1777*x^4/4! + 101541*x^5/5! + 8140411*x^6/6! + 855134533*x^7/7! + 112545136929*x^8/8! + 17984228218057*x^9/9! + ...
such that [x^n] A(x)^(5*n) = (n+4) * [x^(n-1)] A(x)^(5*n) for n>=1.
RELATED SERIES.
A(x)^5 = 1 + 5*x + 35*x^2/2! + 485*x^3/3! + 14545*x^4/4! + 756025*x^5/5! + 57290875*x^6/6! + 5790439625*x^7/7! + 740641270625*x^8/8! + 115751765142125*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(5*n) begins:
n=1: [(1), (5), 35/2, 485/6, 14545/24, 151205/24, ...];
n=2: [1, (10), (60), 1010/3, 6980/3, 21490, 2249000/9, ...];
n=3: [1, 15, (255/2), (1785/2), 51795/8, 449805/8,  ...];
n=4: [1, 20, 220, (5620/3), (44960/3), 389740/3, ...];
n=5: [1, 25, 675/2, 20425/6, (730225/24), (2190675/8), ...];
n=6: [1, 30, 480, 5610, 55980, (534270), (5342700), ...]; ...
in which the coefficients in parenthesis are related by
5 = 5*(1); 60 = 6*(10); 1785/2 = 7*(255/2); 44960/3 = 8*(5620/3); 2190675/8 = 9*(730225/24); 5342700 = 10*(534270); ...
illustrating: [x^n] A(x)^(5*n) = (n+4) * [x^(n-1)] A(x)^(5*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 4*x*A'(x)/A(x)) / (1 - 5*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 7*x^3 + 66*x^4 + 769*x^5 + 10405*x^6 + 157540*x^7 + 2609120*x^8 + 46569365*x^9 + 886686635*x^10 + ... + A300991(n)*x^n + ...

Links

Crossrefs

Cf. A300991, A182962, A300735, A300986, A300988, A300992.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(5*(#A-1))); A[#A] = ((#A+3)*V[#A-1] - V[#A])/(5*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-4*x*A')/(A-5*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 4*x*A'(x)) / (A(x) - 5*x*A'(x)) ).

A300992 E.g.f. A(x) satisfies: [x^n] A(x)^(6*n) = (n+5) * [x^(n-1)] A(x)^(6*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 55, 2233, 141201, 12458731, 1435102663, 206465053425, 35963535971233, 7412714454497491, 1776535156724561751, 488255792062034106793, 152177253891382689328945, 53295007883395937033340603, 20811797234198326671764036071, 9002626614458116653486533691361, 4289501522632944577576478918096193, 2240137918573757743881572713997828515
Offset: 0

Views

Author

Paul D. Hanna, Mar 19 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 55*x^3/3! + 2233*x^4/4! + 141201*x^5/5! + 12458731*x^6/6! + 1435102663*x^7/7! + 206465053425*x^8/8! + 35963535971233*x^9/9! + ...
such that [x^n] A(x)^(6*n) = (n+5) * [x^(n-1)] A(x)^(6*n) for n>=1.
RELATED SERIES.
A(x)^6 = 1 + 6*x + 48*x^2/2! + 720*x^3/3! + 23328*x^4/4! + 1325376*x^5/5! + 109921536*x^6/6! + 12138398208*x^7/7! + 1692740643840*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(6*n) begins:
n=1: [(1), (6), 24, 120, 972, 55224/5, 763344/5, ...];
n=2: [1, (12), (84), 528, 3960, 197568/5, 2494656/5, ...];
n=3: [1, 18, (180), (1440), 11556, 543672/5, 6306336/5, ...];
n=4: [1, 24, 312, (3072), (27648), 1313856/5, 14451264/5, ...];
n=5: [1, 30, 480, 5640, (57420), (574200), 6220080, ...];
n=6: [1, 36, 684, 9360, 107352, (5759424/5), (63353664/5), ...]; ...
in which the coefficients in parenthesis are related by
6 = 6*(1); 84 = 7*(12); 1440 = 8*(180); 27648 = 9*(3072); 574200 = 10*(57420); 63353664/5 = 11*(5759424/5); ...
illustrating: [x^n] A(x)^(6*n) = (n + 5) * [x^(n-1)] A(x)^(6*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 5*x*A'(x)/A(x)) / (1 - 6*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 8*x^3 + 84*x^4 + 1080*x^5 + 16056*x^6 + 266256*x^7 + 4816080*x^8 + 93638016*x^9 + 1937252160*x^10 + ... + A300993(n)*x^n + ...

Links

Crossrefs

Cf. A300993, A182962, A300735, A300986, A300988, A300990, A300992.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(6*(#A-1))); A[#A] = ((#A+4)*V[#A-1] - V[#A])/(6*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-5*x*A')/(A-6*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 5*x*A'(x)) / (A(x) - 6*x*A'(x)) ).

A300596 E.g.f. A(x) satisfies: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4) for n>=1.

Original entry on oeis.org

1, 1, 17, 13171, 56479849, 738706542221, 22885801082965201, 1448479282286023114807, 169382934361790242266135761, 33954915787325983176711221469529, 10997512067125948734754888814957997361, 5482894935903399886164748355296587003210971, 4041251688669102134446309448401146782811371078137
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2018

Keywords

Comments

Compare e.g.f. to: [x^n] exp(x)^(n^4) = n^3 * [x^(n-1)] exp(x)^(n^4) for n>=1.

Examples

			E.g.f.: 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! + 169382934361790242266135761*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^n in A(x)^(n^4) begins:
n=1: [(1), (1), 17/2, 13171/6, 56479849/24, 738706542221/120, ...];
n=2: [1, (16), (256), 113168/3, 114614528/3, 1486010366512/15, ...];
n=3: [1, 81, (7857/2), (636417/2), 1671341283/8, 20586397669407/40, ...];
n=4: [1, 256, 34816, (11641088/3), (2980118528/3), 26464517792512/15, ...];
n=5: [1, 625, 400625/2, 271091875/6, (232095075625/24), (145059422265625/24), ...];
n=6: [1, 1296, 850176, 379068336, 133027474176, (243163666719504/5), (315140112068477184/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 256 = 2^4*16; 636417/2 = 3^4*7857/2; 2980118528/3 = 4^4*11641088/3; ...
illustrating that: [x^n] A(x)^(n^4) = n^4 * [x^(n-1)] A(x)^(n^4).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + 31779658925496*x^6 + 287364845865893467*x^7 + 4200677982722915635200*x^8 + ... + A300597(n)*x^n + ...

Links

Crossrefs

Cf. A182962, A296170, A300590, A300592, A300594, A300597.

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 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300597(n)*x^n, a power series in x with integer coefficients.

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

A300873 E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

Original entry on oeis.org

1, 1, 3, 43, 2041, 197721, 31094251, 7086479443, 2187876597873, 874871971357681, 438740658523346131, 269314248304239932091, 198529013874402868930153, 173067121551267519897494473, 176154202119865662835343738811, 207099741506845262022248534098531, 278645958801870115911315221474653921, 425605862347493892454320041743878801633
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2018

Keywords

Comments

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

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
illustrating that: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ... + A300874(n)*x^n + ...

Links

Crossrefs

Cf. A300874, A300870, A295811, A300590, A296170, A182962.

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] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); EGF=Ser(A); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * n!^2 / n^3, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.75891265... - Vaclav Kotesovec, Aug 11 2021

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.
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