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.

A182962 E.g.f. satisfies: A(x) = exp( x/(1 - x*A'(x)/A(x)) ).

Original entry on oeis.org

1, 1, 3, 25, 433, 12501, 529531, 30495613, 2272643745, 211761416233, 24055076979091, 3267213865097601, 522451410607362193, 97120159467079471165, 20765771676360919883403, 5060640084128464622069221
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 433*x^4/4! +...
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 +...+ A088716(n)*x^(n+1) +...
...
The coefficients of [x^n/n!] in the powers of e.g.f. A(x) begin:
A^1: [(1),(1), 3, 25, 433, 12501, 529531, 30495613, ...];
A^2: [1,(2),(8), 68, 1120, 30832, 1260544, 70737536, ...];
A^3: [1, 3,(15),(135), 2169, 57303, 2261439, 123523515, ...];
A^4: [1, 4, 24,(232),(3712), 94944, 3622336, 192461056, ...];
A^5: [1, 5, 35, 365, (5905),(147625), 5460475, 282185825, ...];
A^6: [1, 6, 48, 540, 8928, (220176),(7926336), 398625408, ...];
A^7: [1, 7, 63, 763, 12985, 318507,(11210479),(549313471), ...];
A^8: [1, 8, 80, 1040, 18304, 449728, 15551104,(743759360), ...];
...
In the above table, the coefficients in parenthesis are related by:
1*1 = 1; 8 = 2^2*2; 135 = 3^2*15; 3712 = 4^2*232; 147625 = 5^2*5905;
this illustrates: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n.
...
Also note that the main diagonal in the above table begins:
[1*1, 2*1, 3*5, 4*58, 5*1181, 6*36696, 7*1601497, 8*92969920, ...];
this illustrates: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n).
...
Let G(x) denote the e.g.f. of A156326:
G(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...
then G(x) satisfies: G(x) = A(x*G(x)) and A(x) = G(x/A(x)) where
G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
...

Links

Crossrefs

Cf. A088716, A300735, A300986, A300988, A300990, A300992.
Cf. A156326, A238223.

Programs

  • Mathematica
    m = 16; A[_] = 1;
Do[A[x_] = Exp[x/(1 - x A'[x]/A[x])] + O[x]^m, {m}];
CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Oct 29 2019 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/(1 - x*deriv(A)/A+x*O(x^n))));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[#A]=((#A-1)*Vec(Ser(A)^(#A-1))[#A-1]-Vec(Ser(A)^(#A-1))[#A])/(#A-1));n!*A[n+1]}

Formula

E.g.f.: A(x) = exp(x*F(x)) where F(x) = 1 + x*F(x)*d/dx[x*F(x)] is the o.g.f. of A088716.
E.g.f. satisfies: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n for n>=1.
E.g.f. satisfies: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n) for n>=0.
E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where A(x*G(x)) = G(x) is the e.g.f. of A156326, which satisfies:
. G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
a(n) ~ c * (n!)^2 * n, where c = 0.21795078944715106549... (see A238223). - Vaclav Kotesovec, Feb 22 2014

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)) ).

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)) ).

A300993 O.g.f. A(x) satisfies: A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).

Original entry on oeis.org

1, 1, 8, 84, 1080, 16056, 266256, 4816080, 93638016, 1937252160, 42339628800, 972303685632, 23365476089856, 585706819083264, 15276194983411200, 413695882240574976, 11612673418376392704, 337392794531354462208, 10133165365696293507072, 314252173854006410465280, 10053170842576476899524608, 331455812860465669006442496
Offset: 1

Views

Author

Paul D. Hanna, Mar 19 2018

Keywords

Comments

O.g.f. equals the logarithm of the e.g.f. of A300992.
The e.g.f. G(x) of A300992 satisfies: [x^n] G(x)^(6*n) = (n+5) * [x^(n-1)] G(x)^(6*n) for n>=1.

Examples

			O.g.f.: 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 + ...
where
A(x) = x * (1 - 5*x*A'(x)) / (1 - 6*x*A'(x)).
RELATED SERIES.
exp(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! + ... + A300992(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 24*x^2 + 336*x^3 + 5400*x^4 + 96336*x^5 + 1863792*x^6 + 38528640*x^7 + 842742144*x^8 + 19372521600*x^9 + ...

Links

Crossrefs

Cf. A300992, A088716, A300736, A300987, A300989, A300991, A300993.

Programs

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

Formula

O.g.f. A(x) satisfies: [x^n] exp( 6*n * A(x) ) = (n + 5) * [x^(n-1)] exp( 6*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^11, where c = 0.00000000002970897246102814... - Vaclav Kotesovec, Mar 20 2018
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