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),", "))
A219230
G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^n)^k) ).
Original entry on oeis.org
1, 1, 2, 5, 13, 32, 82, 201, 498, 1214, 2954, 7117, 17115, 40880, 97336, 230699, 545068, 1283150, 3011783, 7047353, 16445814, 38275172, 88859213, 205796476, 475539242, 1096428621, 2522704211, 5792637135, 13275381694, 30367439045, 69341077367, 158059717986, 359688534284
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 32*x^5 + 82*x^6 + 201*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^2)^3)*...) +
x^3/(3*(1-x^3*(1+x^3))*(1-x^6*(1+x^3)^2)*(1-x^9*(1+x^3)^3)*...) +
x^4/(4*(1-x^4*(1+x^4))*(1-x^8*(1+x^4)^2)*(1-x^12*(1+x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 81*x^5/5 + 228*x^6/6 + 554*x^7/7 + 1399*x^8/8 + 3313*x^9/9 + 7843*x^10/10 +...
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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^m)^k +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), ", "))
A219261
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).
Original entry on oeis.org
1, 1, 3, 9, 33, 124, 503, 2089, 8960, 39142, 173978, 783347, 3567123, 16395199, 75966835, 354447193, 1663921966, 7853325055, 37244059607, 177388171005, 848148206917, 4069483589180, 19588001935380, 94559416543623, 457697632011720, 2220852281129195, 10800560004895426
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 124*x^5 + 503*x^6 + 2089*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^2)^3)*...) +
x^3*A(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*A(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 + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 396*x^5/5 + 1895*x^6/6 + 8989*x^7/7 + 43545*x^8/8 + 211645*x^9/9 + 1036560*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*subst(A, x, x^m +x*O(x^n))/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^k, x, x^m +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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