A107590
G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2).
Original entry on oeis.org
1, 1, 1, 2, 5, 15, 50, 181, 698, 2837, 12062, 53374, 244923, 1162536, 5697119, 28786266, 149814059, 802436166, 4420515689, 25031466730, 145616087486, 869760092469, 5330945435272, 33508699787635, 215863606818041
Offset: 0
A = 1 + x + x^2*A^1 + x^3*A^3 + x^4*A^6 + x^5*A^10 +...
= 1 + x + (x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 15*x^7 +...)
+ (x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 33*x^7 +...)
+ (x^4 + 6*x^5 + 21*x^6 + 62*x^7 +...)
+ (x^5 + 10*x^6 + 55*x^7 +...) +...
= 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 50*x^6 + 181*x^7 +...
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{a(n) = my(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)}
for(n=0,30,print1(a(n),", "))
A155805
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, 3, 19, 191, 2656, 47392, 1034335, 26721781, 798018616, 27058991246, 1027237384009, 43172232488959, 1990253576425960, 99871804451808040, 5419775866582473211, 316301430225674131433, 19756213549154356027408
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 191*x^4/4! + 2656*x^5/5! +...
where e.g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2/2!*A(x)^3 + x^3/3!*A(x)^6 + x^4/4!*A(x)^10 +...
Let B(x) = A(x/B(x)) be the e.g.f. of A155804 then:
B(x) = 1 + x + x^2/2!*B(x) + x^3/3!*B(x)^3 + x^4/4!*B(x)^6 + x^5/5!*B(x)^10 +...
B(x) = 1 + x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 161*x^5/5! + 1606*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)}
A219358
G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n-1)/2).
Original entry on oeis.org
1, 1, 2, 8, 46, 334, 2882, 28604, 320248, 3993184, 54942740, 828086732, 13586200504, 241294019584, 4615319816192, 94629675177320, 2070911506927360, 48185049542009248, 1187816429730925424, 30923773410431125424, 847808674826433774928, 24414218135569507213312
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 334*x^5 + 2882*x^6 +...
where
A(x) = 1 + 1!*x + 2!*x^2*A(x) + 3!*x^3*A(x)^3 + 4!*x^4*A(x)^6 + 5!*x^5*A(x)^10 + 6!*x^6*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.