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 1-6 of 6 results.

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

A300735 E.g.f. A(x) satisfies: [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 31, 697, 25761, 1371691, 97677343, 8869533681, 993709302337, 134086553693011, 21392941696576671, 3977310371182762153, 851537642070562468321, 207892899850805427254907, 57394298500033495294907551, 17789220343418322663802383841, 6151146653207427022767433596033, 2359535664677835451305256629862051, 999033160522078788619730346474821407
Offset: 0

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

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

Examples

			E.g.f.: 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! + ...
such that [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n) for n>=1.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1696*x^4/4! + 60352*x^5/5! + 3134464*x^6/6! + 219316736*x^7/7! + 19655797760*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(2*n) begin:
n=1: [(1), (2), 4, 40/3, 212/3, 7544/15, 195904/45, 13707296/315, ...];
n=2: [1, (4), (12), 128/3, 632/3, 6976/5, 515776/45, 34760896/315, ...];
n=3: [1, 6, (24), (96), 468, 14664/5, 114384/5, 7407552/35, ...];
n=4: [1, 8, 40, (544/3), (2720/3), 82496/15, 1843264/45, 22923136/63, ...];
n=5: [1, 10, 60, 920/3, (4820/3), (9640), 622880/9, 37242080/63, ...];
n=6: [1, 12, 84, 480, 2664, (80448/5), (563136/5), 32495424/35, ...];
n=7: [1, 14, 112, 2128/3, 12572/3, 387128/15, (8018416/45), (64147328/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*(1); 12 = 3*(4); 96 = 4*(24); 2720/3 = 5*(544/3); 9640 = 6*(4820/3); 563136/5 = 7*(80448/5); 64147328/45 = 8*(8018416/45); ...
illustrating that: [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - x*A'(x)/A(x)) / (1 - 2*x*A'(x)/A(x));
explicitly,
log(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 + ...

Links

Crossrefs

Cf. A300736, A182962, A300986, A300988, A300990, A300992.
Cf. A300870, A300590, A296170.

Programs

  • PARI
    {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)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n, A = exp( x*(A-x*A')/(A-2*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) - x*A'(x)) / (A(x) - 2*x*A'(x)) ).
a(n) ~ c * n!^2 * n^3, where c = 0.008789136598... - Vaclav Kotesovec, Oct 24 2020

A300986 E.g.f. A(x) satisfies: [x^n] A(x)^(3*n) = (n + 2) * [x^(n-1)] A(x)^(3*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 37, 1009, 44541, 2799931, 233188033, 24562692897, 3168510747769, 488856473079571, 88597562768075901, 18595324838343722833, 4468203984338696710837, 1217521669261709053889739, 373205252376454629490607641, 127806482596653000272128733761, 48605321514711360780713536416753, 20419150659462692416601828820774307, 9431006202634362924849710001022454869
Offset: 0

Views

Author

Paul D. Hanna, Mar 17 2018

Keywords

Comments

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

Examples

			E.g.f.: 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! + ...
such that [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n) for n>=1.
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2/2! + 171*x^3/3! + 4185*x^4/4! + 173583*x^5/5! + 10491039*x^6/6! + 850141575*x^7/7! + 87745941873*x^8/8! + 11141030530395*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in A(x)^(3*n) begins:
n=1: [(1), (3), 15/2, 57/2, 1395/8, 57861/40, 1165671/80, 18892035/112, ...];
n=2: [1, (6), (24), 102, 576, 21834/5, 206244/5, 15974712/35, ...];
n=3: [1, 9, (99/2), (495/2), 11259/8, 401463/40, 7120899/80, 525246849/560, ...];
n=4: [1, 12, 84, (492), (2952), 102708/5, 864756/5, 60722784/35, ...];
n=5: [1, 15, 255/2, 1725/2, (44595/8), (312165/8), 5077035/16, 340795215/112, ...];
n=6: [1, 18, 180, 1386, 9720, (349542/5), (2796336/5), 36178488/7, ...];
n=7: [1, 21, 483/2, 4179/2, 127323/8, 4767147/40, (76271139/80), (686440251/80), ...]; ...
in which the coefficients in parenthesis are related by
3 = 3*(1); 24 = 4*(6); 495/2 = 5*(99/2); 2952 = 6*(492); 312165/8 = 7*(44595/8); 2796336/5 = 8*(349542/5); 686440251/80 = 9*(76271139/80); ...
illustrating that: [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 2*x*A'(x)/A(x)) / (1 - 3*x*A'(x)/A(x));
explicitly,
log(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 + 36460943208*x^12 + ... + A300987(n)*x^n + ...

Links

Crossrefs

Cf. A300987, A182962, A300735, A300988, A300990, A300992.
Cf. A300870, A300590, A296170.

Programs

  • PARI
    {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)) ); 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-2*x*A')/(A-3*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) - 2*x*A'(x)) / (A(x) - 3*x*A'(x)) ).
a(n) ~ c * (n!)^2 * n^5, where c = 0.0001464056080437... - Vaclav Kotesovec, Mar 20 2018

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)) ).

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), ", "))

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
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