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 1-10 of 11 results. Next

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

A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0) = 1.

Original entry on oeis.org

1, 1, 5, 58, 1181, 36696, 1601497, 92969920, 6908883417, 638746871680, 71860612355981, 9664570175364864, 1531263494465900725, 282321785979644121088, 59935663751282958139425, 14517627118656645274771456, 3980008380007702720451029553, 1226189930561023692489563013120
Offset: 0

Views

Author

Paul D. Hanna, Feb 08 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + ...
log(A(x)) = x + 2^2*x^2/2! + 3^2*5*x^3/3! + 4^2*58*x^4/4! + 5^2*1181*x^5/5! + ...

Links

Crossrefs

Cf. A156325, A156327, A182962, A161968.

Programs

  • Mathematica
    nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = Sum[k^2 * Binomial[n-1,k-1]*b[[k]]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Feb 27 2014 *)
  • PARI
    {a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k^2*a(k-1)*x^k/k!)+x*O(x^n)),n))}
  • PARI
    {a(n)=if(n==0,1,sum(k=1,n,k^2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}
  • PARI
    seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=1, n, k^2 * binomial(n-1,k-1)*a[k]*a[1+n-k])); a} \\ Andrew Howroyd, Jan 05 2020

Formula

a(n) = Sum_{k=1..n} k^2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f.: A(x) = exp( x*A(x) + x^2*A'(x) ). - Paul D. Hanna, Apr 02 2018
E.g.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x/G(x)) = G(x) is the e.g.f. of A182962, which satisfies:
. G(x) = exp( x/(1 - x*G'(x)/G(x)) );
. a(n) = [x^n/n!] G(x)^(n+1)/(n+1) for n>=0.
a(n) = A161968(n+1)/(n+1), where L(x) = x*exp(x*d/dx L(x)) is the e.g.f. of A161968. - Paul D. Hanna, Feb 21 2014
a(n) ~ c * n * (n!)^2, where c = A238223 * exp(1) = 0.592451670452494179138706062417512405957... - Vaclav Kotesovec, Feb 27 2014

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 05 2020

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

A088715 G.f. satisfies: A(x*g(x)) = g(x) where g(x) is the g.f. of A088716.

Original entry on oeis.org

1, 1, 2, 7, 36, 240, 1926, 17815, 184916, 2116498, 26391700, 355405934, 5134778584, 79178537346, 1297633495518, 22522717498167, 412754532495252, 7965288555078018, 161475849044919996, 3431346397643014818
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Links

Crossrefs

Cf. A088716, A238223.

Programs

  • PARI
    a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(k=1,n,(1-x*deriv(log(A)))^(-k)*x^k/k)));polcoeff(A,n) \\ Paul D. Hanna, Aug 31 2009
  • PARI
    a(n)=local(A=1+x); for(i=1, n, A=1+x*A^2/(A-x*deriv(A)+x*O(x^n))); polcoeff(A, n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Mar 20 2013

Formula

G.f.: Coefficient of x^n in A(x)^(n+1)/(n+1) = coefficient of x^n in A(x)^(n+2) = A088716(n).
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/(1 - x*[A'(x)/A(x)])^n/n ). - Paul D. Hanna, Aug 31 2009
G.f. satisfies: A(x) = 1 + x*A(x)^2/(A(x) - x*A'(x)). - Paul D. Hanna, Mar 20 2013
a(n) ~ c * n! * n^2, where c = A238223 / exp(1) = 0.08017961462469262235245081077906956577... - Vaclav Kotesovec, Feb 21 2014

A113668 Self-convolution 8th power of A113674, where a(n) = A113674(n+1)/(n+1).

Original entry on oeis.org

1, 8, 156, 4696, 186406, 9053640, 515875660, 33585910968, 2453913830097, 198609146859416, 17630476159933080, 1703025192274201272, 177846105338917975896, 19968484152350242447288
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2005

Keywords

Comments

From Vaclav Kotesovec, Oct 23 2020: (Start)
In general, for k>=1, if g.f. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^k, then a(n) ~ c(k) * k^n * n! * n^((k-1)/k), where c(k) is a constant (dependent only on k).
c(k) tends to A238223*exp(1) = 0.592451670452494179138706... if k tends to infinity.
(End)

Links

Crossrefs

Cf. A113674, A113662, A113663, A113664, A113665, A113666, A113667, A338377.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=(1+x*deriv(x*A))^8);polcoeff(A,n,x)}

Formula

G.f. satisfies: A(x) = (1 + x*d/dx[x*A(x)] )^8.
a(n) ~ c * 8^n * n! * n^(7/8), where c = 0.6523348263871879460325... - Vaclav Kotesovec, Oct 23 2020

A112915 Recurrence: a(n) = Sum_{k=0..n-1} (k+1)*(n-k)*a(k)*a(n-k-1) for n>0, with a(0)=1.

Original entry on oeis.org

1, 1, 4, 28, 272, 3312, 47872, 794880, 14840064, 306900736, 6953989120, 171200048128, 4548965384192, 129742326218752, 3953689388187648, 128215703582343168, 4409347536459988992, 160304460015345795072
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Links

Crossrefs

Cf. A088716, A112916, A238223.

Programs

  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*(n-k)*a(k)*a(n-k-1)))
  • PARI
    {a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x*deriv(x*F)^2); return(polcoeff(F,n,x))}

Formula

A(x) = 1 + x*G(2*x)^2, where G(x) = g.f. of A088716, such that a(n) = 2^n*A088716(n)/(n+1) for n>=0.
a(n) = 2^(n-1)*A112916(n-1) for n>0.
G.f. satisfies: A(x) = 1 + x*(d/dx[x*A(x)])^2 = 1 + x*(A(x) + x*A'(x))^2.
a(n) ~ c * n * 2^n * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017

A159606 G.f. satisfies: A(x) = 1 + x*d/dx log(1 + x/A(x)).

Original entry on oeis.org

1, 1, -3, 16, -115, 996, -9870, 108816, -1312227, 17116900, -239641798, 3580451040, -56837970358, 955277226736, -16948413979080, 316615678469856, -6213840704926947, 127857371413743540, -2753054722318717950
Offset: 0

Views

Author

Paul D. Hanna, May 16 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 -+...
1/A(x) = 1 - x + 4*x^2 - 23*x^3 + 166*x^4 - 1410*x^5 + 13602*x^6 -+...
log(1+x/A(x)) = x - 3*x^2/2 + 16*x^3/3 - 115*x^4/4 + 996*x^5/5 -+...

Links

Crossrefs

Cf. variants: A159607, A159608.
Cf. A238223.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(log(1+x*Ser(A)^-1)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: x^2*A'(x) = 2*x*A(x) + (1-x)*A(x)^2 - A(x)^3.
a(n) ~ -(-1)^n * c * n! * n^3, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 17 2017

A158884 G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].

Original entry on oeis.org

1, 1, -1, 4, -23, 166, -1410, 13602, -145803, 1711690, -21785618, 298370920, -4372151566, 68234087624, -1129894265272, 19788479904366, -365520041466291, 7103187300763530, -144897616964143050, 3096285550330959336
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
where coefficients in parenthesis form A158883 and signed A088716
and A(x)^-1 (first row) is the g.f. of signed A088715.

Links

Crossrefs

Cf. A158883, A088715, A088716.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);Vec(Ser(A)^(n+1)/(n+1))[n+1]}

Formula

G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017

A338377 G.f. satisfies: A(x) = (1 + x * d/dx(x*A(x)) )^n.

Original entry on oeis.org

1, 1, 9, 226, 10745, 811026, 88058362, 12920344256, 2453913830097, 584608650175630, 170543970449421371, 59769169004510011674, 24775053368568412720967, 11989194513429991057937296, 6698670769128767044654361520, 4280089524780608663200103685056, 3101341801862271814724389007080481
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 23 2020

Keywords

Examples

			a(2) = A113662(2) = 9
a(3) = A113663(3) = 226
a(4) = A113664(4) = 10745
a(5) = A113665(5) = 811026
a(6) = A113666(6) = 88058362
a(7) = A113667(7) = 12920344256
a(8) = A113668(8) = 2453913830097

Links

Crossrefs

Cf. A113662, A113663, A113664, A113665, A113666, A113667, A113668.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=(1+x*deriv(x*A))^n); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = (1 + x * A(x) + x^2 * A'(x) )^n.
a(n) ~ A238223 * exp(1) * n! * n^(n + 1 - 1/n).
a(n) ~ A238223 * exp(1) * n^(n+1) * n! * (1 - log(n)/n).

A193332 E.g.f. satisfies: A(x) = x*exp( A(x)/A'(x) ).

Original entry on oeis.org

1, 2, -3, 52, -1315, 50286, -2655863, 183322952, -15928677063, 1695597280570, -216636191518219, 32688113040335292, -5749136647259226923, 1165789270581830003942, -270019628802455686919295, 70862777375461690495134736, -20921819854506620454336189583
Offset: 1

Views

Author

Paul D. Hanna, Jul 23 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! - 3*x^3/3! + 52*x^4/4! - 1315*x^5/5! + 50286*x^6/6! - 2655863*x^7/7! + 183322952*x^8/8! +...
where A(x)/A'(x) = log(A(x)/x) equals the integer series:
(1) A(x)/A'(x) = x - x^2 + 3*x^3 - 14*x^4 + 85*x^5 - 621*x^6 + 5236*x^7 - 49680*x^8 + 521721*x^9 - 5994155*x^10 +...
which equals -G(-x) where G(x) is the g.f. of A088716.
The series reversion, -L(-x), begins:
(2) -L(-x) = x - 2*x^2/2! + 15*x^3/3! - 232*x^4/4! + 5905*x^5/5! - 220176*x^6/6! + 11210479*x^7/7! - 743759360*x^8/8! +...
where L(x) is the e.g.f. A161968.

Links

Crossrefs

Cf. A088716, A161968, A161967.

Programs

  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(A/(A'+x*O(x^n))));n!*polcoeff(A,n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x)/A'(x) = -G(-x) where G(x) = x + x*G(x)*G'(x) is a g.f. of A088716; thus, log(A(x)/x) is an integer series.
(2) A(-L(-x)) = x where L(x) = x*exp(x*L'(x)) is the e.g.f. of A161968.
a(n) ~ c * (-1)^n * (n!)^2, where c = 0.217950789447151065... (see A238223). - Vaclav Kotesovec, Feb 26 2014
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