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.

A088716 G.f. satisfies: A(x) = 1 + x*A(x)*d/dx[x*A(x)] = 1 + x*A(x)^2 + x^2*A(x)*A'(x).

Original entry on oeis.org

1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Links

Crossrefs

Cf. A112916 (A^2), A112911, A112912, A112913, A112914.
Cf. A088715, A182962, A112915, A218222, A238223.
Cf. A300736, A300987, A300989, A300991, A300993.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(j)*a(n-j-1)*(j+1), j=0..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    a=ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = Sum[k*a[[n-k+1]]*a[[k]],{k,1,n}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 21 2014 *)
m = 20; A[_] = 0;
Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
Map[a,Range[20]] (* Oliver Seipel, Nov 03 2024 ,after Schröder 1870 *)
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*a(k)*a(n-k-1)))
  • PARI
    {a(n)=local(G=1+x);for(i=1,n,G=exp(x/(1 - x*deriv(G)/G+x*O(x^n))));polcoeff(log(G)/x,n)} \\ Paul D. Hanna, Jan 01 2011

Formula

a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.
Forms column 0 of triangle T=A112911, where the matrix inverse satisfies [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.
Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.
G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.
O.g.f.: A(x) = log(G(x))/x where G(x) is the e.g.f. of A182962 given by G(x) = exp( x/(1 - x*G'(x)/G(x)) ). [Paul D. Hanna, Jan 01 2011]
O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) / A(x) = 0 for n>0. - Paul D. Hanna, May 25 2018
O.g.f. A(x) satisfies [x^n] exp( n * x*A(x) ) * (1 - n*x) = 0 for n>0. - Paul D. Hanna, Jul 24 2019
From Paul D. Hanna, Jul 20 2018 (Start):
O.g.f. A(x) satisfies:
* [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^(p+1) * x*A(x)) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0. (End)
a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - Vaclav Kotesovec, Feb 21 2014

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

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
Showing 1-6 of 6 results.

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