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A243161
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G.f. A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - (2*x*F(2,x))^2)^(1/2), F(2,x) = 1/(1 - (3*x*F(3,x))^3)^(1/3), ..., so that F(n-1,x)^n = 1/(1 - (n*x*F(n,x))^n) for n>=0.
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2
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1, 1, 1, 3, 5, 13, 61, 133, 449, 1825, 11497, 29905, 121529, 613121, 3192553, 26963653, 76748369, 367110269, 2101537105, 13742608029, 90490605353, 966603833657, 2899027074937, 15202727310033, 102739122225929, 738145449190921, 6064931145859705, 47996143247509851, 637518525737986877
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OFFSET
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0,4
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 13*x^5 + 61*x^6 + 133*x^7 + 449*x^8 + 1825*x^9 + 11497*x^10 + 29905*x^11 + 121529*x^12 +...
Define F(n,x) with F(0,x) = A(x), where for n>0 we have
F(n-1,x) = 1/(1 - (n*x*F(n,x))^n )^(1/n), or equivalently
F(n,x) = (1 - 1/F(n-1,x)^n)^(1/n) / (n*x),
then
F(1,x) = 1 + 2*x^2 + 3*2*x^4 + 6^2*x^5 + 5*2^2*x^6 + 3^3*2^3*x^7 +...
F(2,x) = 1 + 3^2*x^3 + 6*3^3*x^6 + 12^3*x^7 + 14*3^5*x^9 + 4^4*3^5*x^10 +...
F(3,x) = 1 + 4^3*x^4 + 10*4^5*x^8 + 20^4*x^9 + 30*4^8*x^12 + 5^5*4^7*x^13 +...
F(4,x) = 1 + 5^4*x^5 + 15*5^7*x^10 + 30^5*x^11 + 55*5^11*x^15 + 6^6*5^9*x^16 +...
F(5,x) = 1 + 6^5*x^6 + 21*6^9*x^12 + 42^6*x^13 + 91*6^14*x^18 + 7^7*6^11*x^19 +...
....
F(1,x)^2 = 1 + 2^2*x^2 + 2^4*x^4 + 2^3*3^2*x^5 + 2^6*x^6 + 2*3^2*2^5*x^7 +...
F(2,x)^3 = 1 + 3^3*x^3 + 3^6*x^6 + 3^4*4^3*x^7 + 3^9*x^9 + 2*4^3*3^7*x^10 +...
F(3,x)^4 = 1 + 4^4*x^4 + 4^8*x^8 + 4^5*5^4*x^9 + 4^12*x^12 + 2*5^4*4^9*x^13 +...
F(4,x)^5 = 1 + 5^5*x^5 + 5^10*x^10 + 5^6*6^5*x^11 + 5^15*x^15 + 2*6^5*5^11*x^16 +...
F(5,x)^6 = 1 + 6^6*x^6 + 6^12*x^12 + 6^7*7^6*x^13 + 6^18*x^18 + 2*7^6*6^13*x^19 +...
...
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PROG
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(PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - ((n-k)*x*A)^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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