A218552
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n) ).
Original entry on oeis.org
1, 1, 2, 4, 9, 20, 46, 107, 253, 604, 1463, 3573, 8812, 21901, 54837, 138145, 350068, 891529, 2281092, 5860471, 15113614, 39109461, 101521521, 264286160, 689820642, 1804890193, 4733051924, 12437565725, 32746931264, 86375236835, 228212881032, 603915863737, 1600500761487
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 107*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 117*x^6/6 + 295*x^7/7 + 755*x^8/8 + 1933*x^9/9 + 5048*x^10/10 +...
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*prod(k=1,n\m+1,1+x^(m*k)*subst(A,x,x^k +x*O(x^n))^m))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A218575
G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^k)^n) ).
Original entry on oeis.org
1, 1, 2, 5, 11, 26, 56, 125, 269, 578, 1228, 2600, 5447, 11366, 23575, 48664, 99950, 204383, 416196, 844299, 1706368, 3436555, 6898255, 13803732, 27539833, 54788703, 108703105, 215112006, 424628345, 836218453, 1643005834, 3221104945, 6301628342, 12303151494
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 56*x^6 + 125*x^7 +...
where
log(A(x)) = x/(1*(1-x*(1+x))*(1-x^2*(1+x^2))*(1-x^3*(1+x^3))*...) +
x^2/(2*(1-x^2*(1+x)^2)*(1-x^4*(1+x^2)^2)*(1-x^6*(1+x^3)^2)*...) +
x^3/(3*(1-x^3*(1+x)^3)*(1-x^6*(1+x^2)^3)*(1-x^9*(1+x^3)^3)*...) +
x^4/(4*(1-x^4*(1+x)^4)*(1-x^8*(1+x^2)^4)*(1-x^12*(1+x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 120*x^6/6 + 274*x^7/7 + 527*x^8/8 + 1054*x^9/9 + 1973*x^10/10 + 3807*x^11/11 + 6824*x^12/12 +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,1/(1-x^(m*k)*(1+x^k)^m +x*O(x^n))))),n)}
for(n=0,40,print1(a(n),", "))
A219231
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^n)^k) ).
Original entry on oeis.org
1, 1, 2, 5, 15, 47, 160, 554, 1987, 7243, 26873, 100930, 383412, 1469673, 5679033, 22095308, 86489211, 340360513, 1345814572, 5344184197, 21303295069, 85216434084, 341960332173, 1376212103798, 5553269024152, 22463340663474, 91071265881382, 369996643180885, 1506118767637576
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 47*x^5 + 160*x^6 + 554*x^7 +...
where
log(A(x)) = x/(1*(1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)*...) +
x^2/(2*(1-x^2*A(x^2))*(1-x^4*A(x^2)^2)*(1-x^6*A(x^2)^3)*...) +
x^3/(3*(1-x^3*A(x^3))*(1-x^6*A(x^3)^2)*(1-x^9*A(x^3)^3)*...) +
x^4/(4*(1-x^4*A(x^4))*(1-x^8*A(x^4)^2)*(1-x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 39*x^4/4 + 146*x^5/5 + 594*x^6/6 + 2346*x^7/7 + 9543*x^8/8 + 38710*x^9/9 + 158448*x^10/10 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*prod(k=1, n\m+1, 1/(1-x^(m*k)*subst(A, x, x^m +x*O(x^n))^k))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A219262
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n / Product_{k>=1} (1 - x^(n*k)*A(x^k)^n) ).
Original entry on oeis.org
1, 1, 3, 11, 43, 179, 772, 3446, 15737, 73315, 346869, 1662593, 8055433, 39390631, 194145569, 963490818, 4810435187, 24145458246, 121771909878, 616744946740, 3135657413451, 15997777629900, 81877262767137, 420263466778106, 2162874177974610, 11158378099471260, 57696623896231494
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 772*x^6 + 3446*x^7 +...
where
log(A(x)) = x*A(x)/1/(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
x^2*A(x)^2/2/((1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3/((1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4/((1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 121*x^4/4 + 601*x^5/5 + 2999*x^6/6 + 15247*x^7/7 + 78057*x^8/8 + 403279*x^9/9 + 2095425*x^10/10 +...
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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