A156100 -id:A156100 - cp's OEIS frontend

A156171 G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

1, 1, 3, 11, 53, 357, 3521, 51665, 1122135, 35638903, 1639453459, 108526044099, 10298220348807, 1396920580458279, 270394562069007327, 74574294532698008703, 29276455806256470979269, 16344863466384180848085765, 12969208162308705691408055345, 14616452655308018025267503353697

Offset: 0

Paul D. Hanna, Feb 05 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 53*x^4 + 357*x^5 + 3521*x^6 + 51665*x^7 + 1122135*x^8 + 35638903*x^9 + 1639453459*x^10 + 108526044099*x^11 +...
such that:
log(A(x)) = Sum_{n>=1} x^n/n * (1 + 2^n*x + 4^n*x^2 +...+ 2^(n*k)*x^k +...)^n
or
log(A(x)) = x*(1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 +...) +
x^2/2*(1 + 8*x + 48*x^2 + 256*x^3 + 1280*x^4 + 6144*x^5 +...) +
x^3/3*(1 + 24*x + 384*x^2 + 5120*x^3 + 61440*x^4 + 688128*x^5 +...) +
x^4/4*(1 + 64*x + 2560*x^2 + 81920*x^3 + 2293760*x^4 + 58720256*x^5 +...) +
x^5/5*(1 + 160*x + 15360*x^2 + 1146880*x^3 + 73400320*x^4 + 4227858432*x^5 +...) +
x^6/6*(1 + 384*x + 86016*x^2 + 14680064*x^3 + 2113929216*x^4 + 270582939648*x^5 +...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 161*x^4/4 + 1441*x^5/5 + 18305*x^6/6 + 330625*x^7/7 + 8488961*x^8/8 + 309465601*x^9/9 + 16011372545*x^10/10 + 1174870185985*x^11/11 + 122233833963521*x^12/12 +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A156170, A155200, A156100.

nmax = 20; CoefficientList[Series[Exp[Sum[x^k/(1 - 2^k*x)^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 17 2020 *)

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

a(n) ~ c * 2^(n^2/4 + n + 1/2) / (sqrt(Pi) * n^(3/2)), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.1289368272118771586694585... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.1289312505130275585916134... if n is odd. - Vaclav Kotesovec, Oct 17 2020

A156101 L.g.f.: A(x) = Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} (1 + 2^nx)^n*x^n/n .

Original entry on oeis.org

1, 5, 13, 65, 401, 3521, 43457, 738305, 17746177, 593695745, 27878501377, 1840450134017, 169904883945473, 22139372291866625, 4036405254299041793, 1038968242677362458625, 375102612647535161966593

Offset: 1

Paul D. Hanna, Feb 04 2009

nonn

Compare to l.g.f. Sum_{m>=1} (1 + x)^m * x^m/m of the Fibonacci sequence.

			G.f.: A(x) = x + 5*x^2/2 + 13*x^3/3 + 65*x^4/4 + 401*x^5/5 + ...
A(x) = (1 + 2*x)*x + (1 + 2^2*x)^2*x^2/2 + (1 + 2^3*x)^3*x^3/3 + ...
exp(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 113*x^5 + 741*x^6 + ...

Cf. A156100.

Table[n*Sum[Binomial[n-k,k]*2^(k(n-k))/(n-k),{k,0,Floor[n/2]}],{n,1,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

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

{a(n)=n*sum(k=0,n\2,binomial(n-k,k)*2^(k*(n-k))/(n-k))} \\ Paul D. Hanna, Apr 10 2009

L.g.f.: A(x) = log(G(x)) where G(x) is the g.f. of A156100.

a(n) = n*Sum_{k=0..floor(n/2)} C(n-k,k)*2^(k(n-k))/(n-k). - Paul D. Hanna, Apr 10 2009

Offset corrected by Vaclav Kotesovec, Mar 06 2014

A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)x)^n x^n/n ).

Original entry on oeis.org

1, 1, 2, 5, 14, 40, 159, 812, 5133, 42942, 474619, 6708142, 121367878, 2819170132, 83571532538, 3148951107867, 151069353323782, 9219463980803329, 714951048370178409, 70448496563603216429, 8818161368662624534857

Offset: 0

Paul D. Hanna, Jul 22 2009

nonn

Compare to g.f. of Fibonacci sequence: exp( Sum_{n>=1} A000204(n)*x^n/n ), where A000204 is the Lucas numbers.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 40*x^5 + 159*x^6 +...

Cf. A156216, A156100, A159308, A000204.

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

A163572 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2A006519(n)x)^n * x^n/n ) where A006519(n) is the highest power of 2 dividing n.

Original entry on oeis.org

1, 1, 3, 7, 19, 39, 169, 765, 2183, 4131, 11561, 55157, 666381, 8175433, 68536455, 355280675, 1048740623, 1931107235, 5055100985, 13108206741, 38734589993, 143320957605, 1022112572635, 26523801989399, 914332703582521

Offset: 0

Paul D. Hanna, Jul 31 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 39*x^5 + 169*x^6 +...
log(A(x)) = (1+2*x)*x + (1+4*x)^2*x^2/2 + (1+2*x)^3*x^3/3 + (1+8*x)^4*x^4/4 +...

Cf. A156100, A159308, A163189.

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

cp's OEIS Frontend

A156171 G.f.: A(x) = exp( Sum_{n>=1} x^n/(1 - 2^n*x)^n / n ), a power series in x with integer coefficients.

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A156101 L.g.f.: A(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + 2^n*x)^n*x^n/n .

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A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)*x)^n * x^n/n ).

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A163572 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2*A006519(n)*x)^n * x^n/n ) where A006519(n) is the highest power of 2 dividing n.

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A156101 L.g.f.: A(x) = Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} (1 + 2^nx)^n*x^n/n .

A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)x)^n x^n/n ).

A163572 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2A006519(n)x)^n * x^n/n ) where A006519(n) is the highest power of 2 dividing n.