A218576
G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).
Original entry on oeis.org
1, 1, 2, 4, 7, 14, 25, 44, 79, 137, 237, 408, 689, 1162, 1946, 3231, 5342, 8776, 14340, 23326, 37758, 60847, 97670, 156145, 248697, 394719, 624343, 984360, 1547187, 2424581, 3788730, 5904230, 9176723, 14226914, 22002523, 33947526, 52258177, 80268131, 123028407
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*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 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*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+x^(m*k)*(1+x^k+x*O(x^n))^m )))),n)}
for(n=0,50,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), ", "))
A218551
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^k)^n) ).
Original entry on oeis.org
1, 1, 2, 5, 13, 37, 106, 322, 987, 3119, 9985, 32499, 106910, 355524, 1191960, 4026739, 13689783, 46807685, 160842381, 555175377, 1923970425, 6691769948, 23351250882, 81729943060, 286842588316, 1009256119760, 3559337691300, 12579738946685, 44549347255523, 158058591860684
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 106*x^6 + 322*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 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 342*x^6/6 + 1198*x^7/7 + 4071*x^8/8 + 14356*x^9/9 + 50408*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^k +x*O(x^n))^m)))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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