A107591
G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n+1)/2).
Original entry on oeis.org
1, 1, 2, 6, 22, 91, 408, 1939, 9635, 49614, 263140, 1431301, 7959568, 45152340, 260847526, 1532825675, 9154581802, 55537885743, 342147577227, 2140251570508, 13594688301758, 87702596534110, 574815620158265, 3829029514213952
Offset: 0
A = 1 + x*A^1 + x^2*A^3 + x^3*A^6 + x^4*A^10 + x^5*A^15 ...
= 1 + (x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 91*x^6 +...)
+ (x^2 + 3*x^3 + 9*x^4 + 31*x^5 + 120*x^6 +...)
+ (x^3 + 6*x^4 + 27*x^5 + 116*x^6 +...)
+ (x^4 + 10*x^5 + 65*x^6 +...) +...
= 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 408*x^6 +...
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{a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j*(j+1)/2)+x*O(x^n)));polcoeff(A,n)}
A155804
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n-1)/2).
Original entry on oeis.org
1, 1, 1, 4, 19, 161, 1606, 21022, 323485, 5874913, 122077756, 2871573596, 75437801539, 2193468714373, 70020045331510, 2437979768144026, 92073099488632441, 3753886179551636513, 164556499026975482008
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x + x^2/2!*A(x) + x^3/3!*A(x)^3 + x^4/4!*A(x)^6 + x^5/5!*A(x)^10 +...
Let B(x) = A(x*B(x)) be the e.g.f. of A155805 then:
B(x) = 1 + x*B(x) + x^2/2!*B(x)^3 + x^3/3!*B(x)^6 + x^4/4!*B(x)^10 +...
B(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! + 47392*x^6/6! +...
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{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k*A^(k*(k-1)/2)/k!+x*O(x^n))); n!*polcoeff(A,n)}
A155806
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n^2).
Original entry on oeis.org
1, 1, 3, 22, 269, 4616, 102847, 2824816, 92355769, 3506278528, 151720849691, 7375146930944, 398113181435653, 23640909385071616, 1532325553233566743, 107698939845869111296, 8162300091585206125553, 663836705760309127184384
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4616*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2/2!*A(x)^4 + x^3/3!*A(x)^9 + x^4/4!*A(x)^16 +...
Let B(x) = A(x*B(x)) be the e.g.f. of A155807 then:
B(x) = 1 + x*B(x)^2 + x^2/2!*B(x)^6 + x^3/3!*B(x)^12 + x^4/4!*B(x)^20 +...
B(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23661*x^5/5! + 741013*x^6/6! +...
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{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k*A^(k^2)/k!+x*O(x^n))); n!*polcoeff(A,n)}
A155807
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * A(x)^(n(n+1)).
Original entry on oeis.org
1, 1, 5, 55, 969, 23661, 741013, 28363707, 1284098609, 67149601273, 3984121444581, 264485848799679, 19426332734137849, 1564277403496216293, 137040382838351173301, 12977244383702330201731
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 969*x^4/4! + 23661*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x)^2 + x^2/2!*A(x)^6 + x^3/3!*A(x)^12 + x^4/4!*A(x)^20 +...
Let B(x) = A(x/B(x)) be the e.g.f. of A155806 then:
B(x) = 1 + x*B(x) + x^2/2!*B(x)^4 + x^3/3!*B(x)^9 + x^4/4!*B(x)^16 +...
B(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4616*x^5/5! + 102847*x^6/6! +...
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{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k*A^(k*(k+1))/k!+x*O(x^n))); n!*polcoeff(A,n)}
A219359
G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2).
Original entry on oeis.org
1, 1, 3, 15, 99, 785, 7151, 73097, 827501, 10295549, 140100323, 2077648871, 33466994639, 583535328293, 10971899911703, 221571516903233, 4786126571515961, 110142864494603313, 2690257065904106571, 69500954207702580927, 1893141166438848208747, 54217997416999007587201
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 785*x^5 + 7151*x^6 +...
where
A(x) = 1 + 1!*x*A(x) + 2!*x^2*A(x)^3 + 3!*x^3*A(x)^6 + 4!*x^4*A(x)^10 + 5!*x^5*A(x)^15 +...
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k!*x^k*(A+x*O(x^n))^(k*(k+1)/2)));polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-5 of 5 results.