A155200 -id:A155200 - cp's OEIS frontend

A163138 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n + A(x))^n * x^n/n ).

1, 3, 20, 329, 22584, 7938470, 12605643936, 84977963809781, 2379247465188706528, 273419351336298753589802, 128009562526607810326874017088, 242979581192696030760182903464959706

Offset: 0

Paul D. Hanna, Aug 07 2009

nonn

More generally, we have the following identity:
If A(x,q) = exp( Sum_{n>=1} (q^n + A(x,q))^n * x^n/n ), then
A(x,q) = 1/(1-x*A(x,q))*exp( Sum_{n>=1} q^(n^2)/(1-q^n*x*A(x,q))^n*x^n/n ).
Conjecture: if q is an integer, then A(x,q) is a power series in x with integer coefficients.
Setting q=1 defines the g.f. of the large Schroeder numbers (A006318).

			G.f.: A(x) = 1 + 3*x + 20*x^2 + 329*x^3 + 22584*x^4 + 7938470*x^5 +...
log(A(x)) = [2 + A(x)]*x + [2^2 + A(x)]^2*x^2/2 + [2^3 + A(x)]^3*x^3/3 +...
log(A(x)*(1-xA(x))) = 2/(1-2xA(x))*x + 2^4/(1-4xA(x))^2*x^2/2 + 2^9/(1-8xA(x))^3*x^3/3 +...
log(A(x)) = 3*x + 31*x^2/2 + 834*x^3/3 + 86227*x^4/4 + 39339038*x^5/5 +...

Cf. A202518, A155200.

m = 12; A[] = 1; Do[A[x] = Exp[Sum[(2^n + A[x])^n x^n/n, {n, 1, m}]] + O[x]^m, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)

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

G.f.: A(x) = 1/(1-x*A(x))*exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x*A(x))^n * x^n/n ).

Comment corrected by Paul D. Hanna, Aug 08 2009

A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.

Original entry on oeis.org

1, 2, 5, 26, 501, 42262, 14564184, 18926665052, 96371663657380, 1825266130738144920, 136764680697906838980633, 38133043109557952095731186822, 42464330390232136488003531922964743

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 501*x^4 + 42262*x^5 + ...
log(A(x)) = 2*x + 6*x^2/2 + 56*x^3/3 + 1810*x^4/4 + 206252*x^5/5 + 86874564*x^6/6 + ... + A167010(n)*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A155200, A167006, A167010.

A167010:= func< n | (&+[Binomial(n,j)^n: j in [0..n]]) >;
function A167007(n)
  if n lt 2 then return n+1;
  else return (&+[A167010(j)*A167007(n-j): j in [1..n]])/n;
  end if; return A167007;
end function;
[A167007(n): n in [0..20]]; // G. C. Greubel, Aug 26 2022

A167010[n_]:= A167010[n]= Sum[Binomial[n,j]^n, {j,0,n}];
A167007[n_]:= A167007[n]= If[n==0, 1, (1/n)*Sum[A167010[j]*A167007[n-j], {j,n}]];
Table[A167007[n], {n,0,30}] (* G. C. Greubel, Aug 26 2022 *)

{a(n) = polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^m)*x^m/m) +x*O(x^n)), n)};

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sum(j=0, k, binomial(k, j)^k)*a(n-k)))} \\ Paul D. Hanna, Nov 25 2009

def A167010(n): return sum(binomial(n,j)^n for j in (0..n))
def A167007(n): return 1 if (n==0) else (1/n)*sum( A167010(j)*A167007(n-j) for j in (1..n))
[A167007(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

a(n) = (1/n)*Sum_{k=1..n} A167010(k)*a(n-k) for n>0 with a(0)=1. - Paul D. Hanna, Nov 25 2009

A177399 O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^n * x^n/n ).

Original entry on oeis.org

1, 2, 10, 188, 1414, 53596, 2923652, 44668152, 651967302, 605335444140, 7564881098284, 157357140966472, 96537385644719004, 695895399853879448, 86358988630956719304, 1103071610291574716763120

Offset: 0

Paul D. Hanna, May 30 2010

nonn

Here sigma(n) = A000203(n) is the sum of divisors of n.
Compare g.f. to the formula for Jacobi theta_4(x) given by:
. theta_4(x) = exp( Sum_{n>=1} -(sigma(2n)-sigma(n))*x^n/n )
where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 1414*x^4 + 53596*x^5 +...
log(A(x)) = 2*x + 4^2*x^2/2 + 8^3*x^3/3 + 8^4*x^4/4 + 12^5*x^5/5 +...+ A054785(n)^n*x^n/n +...

Cf. A054785, A000203, A177398, A155200.

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m)-sigma(m))^m*x^m/m)+x*O(x^n)),n)}

A202519 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2A(x) + (-1)^n)^n x^n/n ).

Original entry on oeis.org

1, 1, 7, 27, 165, 877, 5451, 32887, 210505, 1347865, 8859695, 58647219, 393704205, 2662542565, 18166847507, 124738843247, 861922384657, 5986483380145, 41780493605719, 292817777533259, 2060138522838645, 14544377538584925, 103007560370361691, 731635362026777831

Offset: 0

Paul D. Hanna, Dec 22 2011

nonn

			G.f.: A(x) = 1 + x + 7*x^2 + 27*x^3 + 165*x^4 + 877*x^5 + 5451*x^6 +...
where
log(A(x)) = (2*A(x) - 1)*x + (2*A(x) + 1)^2*x^2/2 + (2*A(x) - 1)^3*x^3/3 + (2*A(x) + 1)^4*x^4/4 +...
log(A(x)*(1-2*x*A(x))) = -1/(1 + 2*x*A(x))*x + 1/(1 - 2*x*A(x))^2*x^2/2 - 1/(1 + 2*x*A(x))^3*x^3/3 + 1/(1 - 2*x*A(x))^4*x^4/4 +...

Cf. A185385, A163138, A202669, A155200.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2*A+(-1)^m+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}

G.f. satisfies: A(x) = 1/(1-2*x*A(x)) * exp( Sum_{n>=1} (-1)^n/(1 - (-1)^n*2*x*A(x))^n * x^n/n ).
G.f. satisfies: A(x) = sqrt( (1 - (2*A(x)-1)^2*x^2)/(1 - (2*A(x)+1)^2*x^2) ) / (1 - (2*A(x)-1)*x).
G.f. satisfies: 0 = -(1-x) - 2*x*A(x) + (1-x)*(1+x)^2*A(x)^2 - 2*x*(1+x)^2*A(x)^3 - 2^2*x^2*(1-x)*A(x)^4 + 2^3*x^3*A(x)^5.

A202668 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 2, 4, 12, 42, 158, 618, 2498, 10360, 43832, 188420, 820608, 3613212, 16057640, 71933768, 324482500, 1472604586, 6719100254, 30804229858, 141829955338, 655541387406, 3040527731790, 14147444737654, 66018910398574, 308898542610666, 1448867831911170

Offset: 0

Paul D. Hanna, Dec 22 2011

nonn

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 158*x^5 + 618*x^6 + ...
where
log(A(x)) = (A(x) + 1)*x + (A(x) - 1)^2*x^2/2 + (A(x) + 1)^3*x^3/3 + (A(x) - 1)^4*x^4/4 + ...
log( A(x)*(1-x*A(x)) ) = 1/(1 + x*A(x))*x + 1/(1 - x*A(x))^2*x^2/2 + 1/(1 + x*A(x))^3*x^3/3 + 1/(1 - x*A(x))^4*x^4/4 + ...
From _Paul D. Hanna_, Oct 11 2024: (Start)
SPECIFIC VALUES.
A(t) = 2 at t = 0.195782060076367892865630673522992184838101...
where 12*t^3 - 4*t^2 - 15*t + 3 = 0.
A(t) = 3/2 at t = 0.1528468026979892250300352740045422934687...
where 45*t^3 - 18*t^2 - 260*t + 40 = 0.
A(1/6) = 1.5975588141693553913621853542774164447766461118908...
A(1/7) = 1.4422077780342017637064340698606478883307441400444...
A(1/8) = 1.3558965312086216338851741626422486193364696459775...
A(1/9) = 1.2992876417963412242026519185070094965390617289384...
A(1/10) = 1.258828814568496961617240364573696812116531654741...
(End)

Paul D. Hanna, Table of n, a(n) for n = 0..800
Vaclav Kotesovec, Recurrence (of order 12)

Cf. A202669, A185385, A202630, A202518, A155200.

{a(n) = my(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (A - (-1)^m +x*O(x^n))^m * x^m/m))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

G.f. satisfies: A(x) = 1/(1-x*A(x)) * exp( Sum_{n>=1} 1/(1 - (-1)^n*x*A(x))^n * x^n/n ).
G.f. satisfies: A(x) = sqrt( (1 - (A(x)+1)^2*x^2)/(1 - (A(x)-1)^2*x^2) ) / (1 - (A(x)+1)*x).
G.f. satisfies: 0 = -(1+x) - x*A(x) + (1+x)*(1-x)^2*A(x)^2 - x*(1-x)^2*A(x)^3 - x^2*(1+x)*A(x)^4 + x^3*A(x)^5.
From Vaclav Kotesovec, Oct 11 2024: (Start)
a(n) ~ sqrt((-1 - s + (-1 - 2*r + 3*r^2)*s^2 + (-1 + 4*r - 3*r^2)*s^3 - r*(2 + 3*r)*s^4 + 3*r^2*s^5)/(1 - r*(1 + 3*s) + r^2*(-1 + 6*s - 6*s^2) + r^3*(1 - 3*s - 6*s^2 + 10*s^3))) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.20089689587759865228481815120918189691453519374477284069915... and s = 2.3487742728380350386577466365052703249852809669846393564277... are positive real roots of the system of equations s^2*(1 + r^3*(-1 + s)^2*(1 + s)) = 1 + r^2*(-1 + s)^2*s^2 + r*(1 + s + s^2 + s^3) and 2*(-1 + r)^2*(1 + r)*s + 5*r^3*s^4 = r*(1 + 3*(-1 + r)^2*s^2 + 4*r*(1 + r)*s^3).
Numerically, a(n) ~ c * d^n / n^(3/2), where d = 1/r = 4.977677706923229216140896605827075562322447814212438341196056039... and c = 0.7100736662419384614471705442776864037581200760804364785319... (End)

A156214 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)(xA(x))^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 14, 256, 18734, 6932928, 11550075900, 80606017093632, 2307293302418365718, 268696321569450570148864, 126770971088210751226430473604, 241680859880056839468193961216049152

Offset: 0

Paul D. Hanna, Feb 06 2009

nonn

Compare to g.f. for Catalan sequence: C(x) = exp( Sum_{n>=1} (x*C(x))^n/n ).

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 256*x^3 + 18734*x^4 + 6932928*x^5 +...
log(A(x)) = 2*x + 24*x^2/2 + 692*x^3/3 + 72704*x^4/4 + 34465932*x^5/5 +...
log(A(x)) = 2*xA(x) + 2^4*(xA(x))^2/2 + 2^9*(xA(x))^3/3 + 2^16*(xA(x))^4/4 + ...

Cf. A000108, A156213.

terms = 12;
g[n_] := g[n] = If[n == 0, 1, (1/n)*Sum[2^(k^2)*g[n - k], {k, 1, n}]];
G[x_] = Sum[g[n]*x^n, {n, 0, terms}];
A[x_] = (1/x)*InverseSeries[Series[x/G[x], {x, 0, terms}], x];
CoefficientList[A[x] + O[x]^terms, x] (* Jean-François Alcover, Nov 14 2017 *)

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

G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x*G(x)) = G(x) is the g.f. of A155200. [Paul D. Hanna, Jun 30 2009]

A156335 G.f.: A(x) = exp( Sum_{n>=1} 2^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 4, 16, 92, 1816, 47344, 4888640, 546663016, 245429690704, 113080892367776, 209848258185362560, 393950238751186551328, 2976605303522286162203456, 22642571073509592590956639360, 692351532949951721637759480882688

Offset: 0

Paul D. Hanna, Feb 10 2009

nonn

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 92*x^4 + 1816*x^5 + 47344*x^6 +...
log(A(x)) = 2*x + 2^2*x^2/2 + 2^5*x^3/3 + 2^8*x^4/4 + 2^13*x^5/5 + 2^18*x^6/6 +...

Cf. A156334, A156336, A156337, A155200.

{a(n)=polcoeff(exp(sum(k=1, n, 2^floor((k^2+1)/2)*x^k/k)+x*O(x^n)), n)}

a(n) = (1/n)*Sum_{k=1..n} 2^floor((k^2+1)/2) * a(n-k) for n>0, with a(0)=1.

A156337 G.f.: A(x) = exp( Sum_{n>=1} 4^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 16, 384, 17856, 13492992, 11507268608, 160888878129152, 2306486569154275328, 537309590223329223155712, 126767209261235580163634135040, 483356141899716284828508078471905280

Offset: 0

Paul D. Hanna, Feb 10 2009

nonn

It appears that g.f. exp( Sum_{n>=1} m^[(n^2+1)/2]*x^n/n ) forms a power series in x with integer coefficients for any positive integer m.

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 384*x^3 + 17856*x^4 + 13492992*x^5 +...
log(A(x)) = 4*x + 4^2*x^2/2 + 4^5*x^3/3 + 4^8*x^4/4 + 4^13*x^5/5 + 4^18*x^6/6 +...

Cf. A156335, A156336, A155207, A155200.

{a(n)=polcoeff(exp(sum(k=1, n, 4^floor((k^2+1)/2)*x^k/k)+x*O(x^n)), n)}

a(n) = (1/n)*Sum_{k=1..n} 4^floor((k^2+1)/2) * a(n-k) for n>0, with a(0)=1.

A156340 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2-n+1) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 2, 6, 52, 2150, 423804, 358766428, 1257303170984, 18016913850523398, 1049450810327077624300, 247590106794776589832254260, 236013988752078034604114551553880, 907420117150975291421488593816623266780, 14052902173791695936955751957273562543384799320

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 52*x^3 + 2150*x^4 + 423804*x^5 + ...
log(A(x)) = 2*x + 2^3*x^2/2 + 2^7*x^3/3 + 2^13*x^4/4 + 2^21*x^5/5 + 2^31*x^6/6 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A155200.

{a(n)=polcoeff(exp(sum(k=1,n,2^(k^2-k+1)*x^k/k)+x*O(x^n)),n)}
for(n=0,15,print1(a(n),", "))

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,2^(k^2-k+1)*a(n-k)))}
for(n=0,15,print1(a(n),", "))

a(n) = (1/n)*Sum_{k=1..n} 2^(k^2-k+1) * a(n-k) for n>0, with a(0)=1.

a(n) ~ 2^(n^2 - n + 1) / n. - Vaclav Kotesovec, Oct 07 2020

Terms a(12) and beyond from Andrew Howroyd, Jan 05 2020

A157315 G.f.: A(x) = sin( Sum_{n>=0} 2^((2n+1)^2) * C(2n,n)/4^n * x^(2n+1)/(2n+1) ); alternating zeros omitted.

Original entry on oeis.org

2, 84, 2516412, 25131689308776, 73459034127708442263660, 59475400379433834763260101514326040, 12984879931670595437855043594849682375333268239320

Offset: 1

Paul D. Hanna, Mar 17 2009

nonn

Compare g.f. to the expansion of the inverse sine of x:

arcsin(x) = Sum_{n>=0} C(2n,n)/4^n * x^(2n+1)/(2n+1).

			G.f.: A(x) = 2*x + 84*x^3 + 2516412*x^5 + 25131689308776*x^7 + ...
The inverse sine of A(x) begins:
arcsin(A(x)) = 2*x + 2^9*(2/4)*x^3/3 + 2^25*(6/4^2)*x^5/5 + 2^49*(20/4^3)*x^7/7 + 2^81*(70/4^4)*x^9/9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..29

Cf. A000984 (C(2n, n)), A136558, A155200.

m:=30;
R:=PowerSeriesRing(Rationals(), m);
b:=Coefficients(R!( Sin( (&+[2^(4*j^2+2*j+1)*Binomial(2*j,j)*x^(2*j+1)/(2*j+1): j in [0..m+2]]) ) ));
[b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Mar 16 2021

m := 30;
S := series( sin(add(2^(4*j^2+2*j+1)*binomial(2*j,j)*x^(2*j+1)/(2*j+1), j = 0..m+2)), x, m+1);
seq(coeff(S, x, 2*j+1), j = 0..m/2); # G. C. Greubel, Mar 16 2021

With[{m = 30}, CoefficientList[Series[Sin[Sum[2^(4*n^2+2*n+1)*((n+1)/(2*n+1)) *CatalanNumber[n]*x^(2*n+1), {n,0,m+2}]], {x,0,m}], x]][[2 ;; ;; 2 ]] (* G. C. Greubel, Mar 16 2021 *)

{a(n)=polcoeff(sin(sum(m=0,n\2,2^((2*m+1)^2)*binomial(2*m,m)/4^m*x^(2*m+1)/(2*m+1))+x*O(x^n)),n)}

m=30
def A157315_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sin( sum(2^(4*j^2+2*j+1)*binomial(2*j,j)*x^(2*j+1)/(2*j+1) for j in [0..m+2])) ).list()
a=A157315_list(m); [a[2*n+1] for n in (0..(m-2)/2)] # G. C. Greubel, Mar 16 2021

cp's OEIS Frontend

A163138 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n + A(x))^n * x^n/n ).

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A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.

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A177399 O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^n * x^n/n ).

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A202519 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) + (-1)^n)^n * x^n/n ).

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A202668 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) - (-1)^n)^n * x^n/n ).

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A156214 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)*(x*A(x))^n/n ), a power series in x with integer coefficients.

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A156335 G.f.: A(x) = exp( Sum_{n>=1} 2^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

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A156337 G.f.: A(x) = exp( Sum_{n>=1} 4^[(n^2+1)/2]*x^n/n ), a power series in x with integer coefficients.

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A202519 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2A(x) + (-1)^n)^n x^n/n ).

A156214 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)(xA(x))^n/n ), a power series in x with integer coefficients.