A110448
G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).
Original entry on oeis.org
1, 1, 2, 3, 6, 8, 18, 23, 49, 73, 145, 194, 474, 611, 1331, 2027, 4393, 5919, 14736, 19415, 46487, 68504, 156618, 212055, 560380, 739165, 1833012, 2657837, 6513367, 8743208, 23649777, 31140300, 81276046, 114962333, 293600318, 391926154
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...
where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or
A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).
The g.f. can also be expressed as the product:
A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...
where the functions are g.f.s of well-known sequences:
G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;
G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;
G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;
G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.
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{a(n)=polcoeff(exp(x*Ser(vector(n,m, sumdiv(m,d,binomial(m,d))/m))+x*O(x^n)),n)}
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{a(n)=polcoeff(prod(m=1,n,1/x*serreverse(x/(1+x^m +x*O(x^n)))),n)}
A206289
G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).
Original entry on oeis.org
1, 1, 2, 4, 10, 25, 73, 214, 679, 2189, 7331, 24867, 86269, 302144, 1072621, 3837768, 13853674, 50319789, 183941789, 675731105, 2494370326, 9244865453, 34394851701, 128390336942, 480749791772, 1805161153783, 6795744287172, 25643914891284, 96980809856731
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x*(1 - x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
Note that G_n(x) = x + x*G_n(x)^(n+1).
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{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
for(n=0,35,print1(a(n),", "))
A206293
G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/A(x^k) ).
Original entry on oeis.org
1, 1, 2, 5, 18, 78, 415, 2467, 16212, 114623, 863229, 6858780, 57156213, 497147291, 4497291265, 42189445764, 409478828567, 4103901097024, 42403116824997, 451059832858894, 4933844398096693, 55436157047213427, 639215949145395559, 7557505365363885063
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 78*x^5 + 415*x^6 + 2467*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n(x) satisfies: G_n( x/A(x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 3*x^3 + 12*x^4 + 59*x^5 + 329*x^6 + 2035*x^7 +...
G_2(x) = x + x^3 + 4*x^5 + 22*x^7 + 144*x^9 + 1045*x^11 + 8159*x^13 +...
G_3(x) = x + x^4 + 5*x^7 + 35*x^10 + 289*x^13 + 2626*x^16 +...
G_4(x) = x + x^5 + 6*x^9 + 51*x^13 + 510*x^17 + 5597*x^21 +...
G_5(x) = x + x^6 + 7*x^11 + 70*x^16 + 823*x^21 + 10608*x^26 +...
G_6(x) = x + x^7 + 8*x^13 + 92*x^19 + 1244*x^25 + 18434*x^31 +...
G_7(x) = x + x^8 + 9*x^15 + 117*x^22 + 1789*x^29 + 29975*x^36 +...
where G_n(x) = x*A( G_n(x)^n ).
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{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,prod(k=1,m,serreverse(x/subst(A,x,x^k +x*O(x^n))))));polcoeff(A,n)}
for(n=0,45,print1(a(n),", "))
Showing 1-3 of 3 results.
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