A219229
G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).
Original entry on oeis.org
1, 1, 2, 4, 9, 20, 45, 97, 217, 476, 1043, 2256, 4890, 10540, 22695, 48614, 103856, 221103, 469738, 995299, 2104514, 4439410, 9345471, 19631897, 41162937, 86148416, 179987656, 375411910, 781773332, 1625473202, 3374684346, 6996168098, 14483902403, 29945326755
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 97*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 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 111*x^6/6 + 232*x^7/7 + 555*x^8/8 + 1204*x^9/9 + 2608*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^m+x*O(x^n))^k )))), n)}
for(n=0, 50, 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),", "))
A219232
G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).
Original entry on oeis.org
1, 1, 2, 4, 11, 30, 92, 284, 918, 3005, 10043, 33943, 116138, 400862, 1395228, 4889389, 17240482, 61117789, 217709832, 778841527, 2797066886, 10080379573, 36444817306, 132147553180, 480444008087, 1751033068088, 6396352141777, 23414462628460, 85878613308907, 315556155264918
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 30*x^5 + 92*x^6 + 284*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 + 7*x^3/3 + 27*x^4/4 + 86*x^5/5 + 321*x^6/6 + 1128*x^7/7 + 4163*x^8/8 + 15172*x^9/9 + 56078*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^m +x*O(x^n))^k)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A219263
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, 10, 39, 159, 693, 3101, 14292, 67116, 320448, 1549834, 7579037, 37406737, 186102602, 932294987, 4698796087, 23809155711, 121219100012, 619800529792, 3181291257740, 16385813881342, 84666104373097, 438742341955132, 2279628504172080, 11873579440176774, 61984238371422197
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 159*x^5 + 693*x^6 + 3101*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 + 22*x^3/3 + 109*x^4/4 + 531*x^5/5 + 2726*x^6/6 + 13952*x^7/7 + 72581*x^8/8 + 379264*x^9/9 + 1994875*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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