A096538 -id:A096538 - cp's OEIS frontend

A141369 E.g.f. satisfies A(x) = exp(x*A(-x)).

1, 1, -1, -8, 21, 336, -1445, -35328, 212009, 7010560, -54073449, -2258780160, 21303275389, 1076400869376, -12005345614093, -712084337721344, 9169911825026385, 624667885401341952, -9122376282532978769, -701910552416102645760, 11462725659070874233061

Offset: 0

Paul D. Hanna, Jun 28 2008

sign

			E.g.f.: A(x) = 1 + x - x^2/2! - 8*x^3/3! + 21*x^4/4! + 336*x^5/5! --++ ...
Log(A(x)) = x - x^2 - x^3/2! + 8*x^4/3! + 21*x^4/4! - 336*x^5/5! -++- ...

Seiichi Manyama, Table of n, a(n) for n = 0..405

Cf. A096538.

Flatten[{1,Table[Sum[(-1)^(n-k) * Binomial[n,k] * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n)=local(A=1); for(i=0, n, A=exp((-1)^(n-i)*x*A+x*O(x^n))); n!*polcoeff(A, n)}

{a(n)=sum(k=0,n,(-1)^(n-k)*binomial(n,k)*(n-k+1)^(k-1)*k^(n-k))} \\ Paul D. Hanna, Jun 13 2009

E.g.f.: A(x) = exp(x*exp(-x*exp(x*exp(-x*exp(x*...))))).
a(n+1) = Sum_{i=0..n} (i+1)*(-1)^i*binomial(n,i)*a(i)*a(n-i) - from a formula given in A096538 by Vladeta Jovovic.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (n-k+1)^(k-1) * k^(n-k). - Paul D. Hanna, Jun 13 2009
|a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.63217617290426743984700577681768332... if n is even, and c = 1.4315233793609300008688492299361204... if n is odd. - Vaclav Kotesovec, Feb 26 2014

A385526 E.g.f. A(x) satisfies A(x) = exp(xA(3x)).

Original entry on oeis.org

1, 1, 7, 208, 23365, 9588976, 14040296659, 71747056999360, 1255862559932597257, 74168744207577385109248, 14599375893944236344767578111, 9483024632344097320792984610415616, 20158786175666520486280070249843236771213, 139271933359690469686747131442731382830399594496

Offset: 0

Seiichi Manyama, Jul 02 2025

nonn

Cf. A000272, A096538, A141369, A385527, A385528, A385529, A385530.

Cf. A015084.

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[3*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385526(n)
  A(3, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 3^k * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 3^(n*(n-1)/2), where c = 1.361839192264541770366149558100099215697354561... - Vaclav Kotesovec, Jul 02 2025

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).

Original entry on oeis.org

1, 1, 9, 457, 118961, 152894961, 940318147705, 26967408304580857, 3534888068831469959649, 2084993641133372935803249505, 5465706581663919414225671125834601, 63043356313898446097762231466174924913065, 3173076775252515207774429654590479617164788572049

Offset: 0

Seiichi Manyama, Jul 02 2025

nonn

Cf. A000272, A096538, A141369, A385526, A385528, A385529, A385530.

Cf. A015085.

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[4*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385527(n)
  A(4, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 4^k * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 2^(n*(n-1)), where c = 1.216702003338638031273833889488221691367428313263423339843... - Vaclav Kotesovec, Jul 02 2025

A385528 E.g.f. A(x) satisfies A(x) = exp(xA(-2x)).

Original entry on oeis.org

1, 1, -3, -47, 1385, 119601, -22345691, -10181013695, 10346973518097, 23934447308323873, -122307331801326167539, -1379021793666951568998159, 33874331587448813081748999673, 1804181313330860398948564389193681, -206892703326367302570264123699846971211

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385529, A385530.

Cf. A015097.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385528(n)
  A(-2, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-2)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A385529 E.g.f. A(x) satisfies A(x) = exp(xA(-3x)).

Original entry on oeis.org

1, 1, -5, -152, 15949, 6548656, -9510189137, -48598095401792, 849885323784381337, 50192539805114962349824, -9878895951508580401879879229, -6416836884643090722807370469927936, 13640603845766595275775514993987722683941, 94239467260528503337471761892783659993298198528

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385530.

Cf. A015098.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385529(n)
  A(-3, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-3)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A385530 E.g.f. A(x) satisfies A(x) = exp(xA(-4x)).

Original entry on oeis.org

1, 1, -7, -359, 90705, 116586321, -715618113143, -20523234900205911, 2689857437569063003169, 1586566688643256394542888225, -4159073515698730238218108546470759, -47972197129364591236078587520718376138951, 2414519164037893898620724924577882591112859773297

Offset: 0

Seiichi Manyama, Jul 02 2025

sign

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385529.

Cf. A015099.

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385530(n)
  A(-4, n)
end

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-4)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A168600 E.g.f. A(x) satisfies A(x) = exp( xA(2x)^2 ).

Original entry on oeis.org

1, 1, 9, 265, 19889, 3506801, 1417530745, 1302573091513, 2700478102745057, 12518436654808255585, 128568477648089286062441, 2900655737241126221237790185, 142677722979145454671155940121233, 15200178301599487957128451391538504145

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

			E.g.f: A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 19889*x^4/4! +...

Robert Israel, Table of n, a(n) for n = 0..76

Cf. A096538, A168601.

Cf. A360987.

F:= A -> A(x) - exp(x*A(2*x)^2):
Extend:= proc(ff)
  local f1x, m, f2, S, R, i;
  f1x:= ff(x); m:= degree(f1x,x);
  f2:= unapply(ff(x) + add(a[i]*x^i,i=m+1..2*m+1),x);
  S:= series(F(f2),x,2*m+2);
  R:= solve(identity(convert(S,polynom),x),{seq(a[i],i=m+1..2*m+1)});
  unapply(subs(R, f2(x)),x);
end proc:
g:= 1:
for iter from 1 to 5 do g:= Extend(g) od:
seq(coeff(g(x),x,j)*j!,j=0..31); # Robert Israel, Feb 22 2019

nmax = 13; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Exp[x A[2 x]^2] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k n!];
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(x*subst(A, x, 2*x)^2) ); n!*polcoeff(A, n)}

a(n) ~ c * n! * 2^(n*(n+1)/2), where c = 0.986274628764911276343959993... - Vaclav Kotesovec, Jul 02 2025

a(0) = 1; a(n) = 2^(n-1) * (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (n-i)/2^i * a(i) * a(j) * a(k)/(i! * j! * k!). - Seiichi Manyama, Jul 06 2025

A168601 E.g.f. A(x) satisfies A(x) = exp( xA(2x)^3 ).

Original entry on oeis.org

1, 1, 13, 577, 65737, 17553361, 10704000277, 14796719274961, 46078793355045073, 320622833897384770657, 4940987529761176087132381, 167239138427197351352909547169, 12340269209872740741602975099855641

Offset: 0

Paul D. Hanna, Dec 05 2009

nonn

			E.g.f: A(x) = 1 + x + 13*x^2/2! + 577*x^3/3! + 65737*x^4/4! +...

Cf. A096538, A168600.

Cf. A360988.

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(x*subst(A, x, 2*x)^3) ); n!*polcoeff(A, n)}

a(0) = 1; a(n) = 2^(n-1) * (n-1)! * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (n-i)/2^i * a(i) * a(j) * a(k) * a(l)/(i! * j! * k! * l!). - Seiichi Manyama, Jul 06 2025

A196734 E.g.f. satisfies: A(x) = exp(xA(2x)^(1/2)).

Original entry on oeis.org

1, 1, 3, 22, 377, 15236, 1458577, 326046946, 166826961233, 192154584592072, 491898410990385281, 2770349200953966300494, 34041983929934523771795481, 906333341309409985333411618492, 51972772881917637838407651811301201, 6386414140694907598544170345261596881026

Offset: 0

Paul D. Hanna, Oct 05 2011

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 377*x^4/4! + 15236*x^5/5! +...
where
A(2*x)^(1/2) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2649*x^4/4! + 226881*x^5/5! + 45061213*x^6/6! +...+ A096538(n)*x^n/n! +...

Cf. A096538, A196735.

{a(n)=local(A=1+x);for(i=1,21,A=exp(x*subst(A,x,2*x+x*O(x^n))^(1/2)));n!*polcoeff(A,n)}

E.g.f.: A(x) = G(x/2)^2 where G(x) = e.g.f. of A096538.

A385620 E.g.f. A(x) satisfies A(x) = exp( x(A(2x) + A(3*x)) ).

Original entry on oeis.org

1, 2, 24, 1064, 158144, 78427712, 130391102464, 725657074158592, 13450842239318679552, 825492067428121929359360, 166724642619378284453845213184, 110175812687250637947409895640473600, 236918101449618886434191300434062010777600, 1649425480856495624442166311045759714226010423296

Offset: 0

Seiichi Manyama, Jul 05 2025

nonn

Cf. A058014, A096538, A385526.

terms = 14; A[] = 1; Do[A[x] =Exp[x*(A[2*x] + A[3*x])]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 05 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (j+1)*(2^j+3^j)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (2^k+3^k) * binomial(n-1,k) * a(k) * a(n-1-k).

cp's OEIS Frontend

A141369 E.g.f. satisfies A(x) = exp(x*A(-x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Ruby

Formula

A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Ruby

Formula

A385528 E.g.f. A(x) satisfies A(x) = exp(x*A(-2*x)).

Views

Author

Keywords

Crossrefs

Programs

Ruby

Formula

A385529 E.g.f. A(x) satisfies A(x) = exp(x*A(-3*x)).

Views

Author

Keywords

Crossrefs

Programs

Ruby

Formula

A385530 E.g.f. A(x) satisfies A(x) = exp(x*A(-4*x)).

Views

Author

Keywords

Crossrefs

Programs

Ruby

Formula

A168600 E.g.f. A(x) satisfies A(x) = exp( x*A(2*x)^2 ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A168601 E.g.f. A(x) satisfies A(x) = exp( x*A(2*x)^3 ).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A196734 E.g.f. satisfies: A(x) = exp(x*A(2*x)^(1/2)).

Views

Author

Keywords

A385526 E.g.f. A(x) satisfies A(x) = exp(xA(3x)).

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).

A385528 E.g.f. A(x) satisfies A(x) = exp(xA(-2x)).

A385529 E.g.f. A(x) satisfies A(x) = exp(xA(-3x)).

A385530 E.g.f. A(x) satisfies A(x) = exp(xA(-4x)).

A168600 E.g.f. A(x) satisfies A(x) = exp( xA(2x)^2 ).

A168601 E.g.f. A(x) satisfies A(x) = exp( xA(2x)^3 ).

A196734 E.g.f. satisfies: A(x) = exp(xA(2x)^(1/2)).

A385620 E.g.f. A(x) satisfies A(x) = exp( x(A(2x) + A(3*x)) ).