A090753
Coefficients of power series A(x) such that n-th term of A(x)^n = n!*n*x^(n-1), for n>0.
Original entry on oeis.org
1, 2, 2, 4, 16, 88, 600, 4800, 43680, 443296, 4949920, 60217408, 792134528, 11200176128, 169375195136, 2728019576832, 46626359376384, 842947307334144, 16073131554826752, 322403473258650624, 6786861273524305920
Offset: 0
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a(n)=if(n<0,0,polcoeff(x/serreverse(sum(k=1,n+1,k!*x^k,x^2*O(x^n))),n)) /* Michael Somos, Feb 14 2004 */
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a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m/(A+m*x+x*O(x^n))^m)); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2013
A136633
G.f.: A(x) = Series_Reversion( x / Sum_{n>=0} (n+1)!*x^n ).
Original entry on oeis.org
1, 2, 10, 68, 544, 4832, 46312, 471536, 5055328, 56795840, 667286656, 8197599104, 105446118784, 1423627264256, 20234885027968, 303737480337152, 4827671316780544, 81385455480335360, 1455806861755411456
Offset: 0
G.f.: A(x) = 1 + 2*x + 10*x^2 + 68*x^3 + 544*x^4 + 4832*x^5 + 46312*x^6 +...
Let F(x) = 1 + 2x + 6x^2 + 24x^3 + 120x^4 + 720x^5 +...+ (n+1)!*x^n +...
then A(x) = F(x*A(x)) and A(x/F(x)) = F(x);
also, a(n) = coefficient of x^n in F(x)^n divided by (n+1).
The g.f. A(x) also satisfies:
A(x) = 1 + 2*x*A(x)/(1+x*A(x)) + 2*2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 2*3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 2*4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +...
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a(n)=polcoeff(serreverse(x/sum(k=0,n,(k+1)!*x^k +x*O(x^n))),n)
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a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m*A^m/(1+m*x*A+x*O(x^n))^m)); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2013
A222012
G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(x)^n / (1 + n*x*A(x))^n.
Original entry on oeis.org
1, 1, 4, 22, 145, 1081, 8863, 78751, 752587, 7708483, 84532222, 992628616, 12487788067, 168344145919, 2430351826084, 37517872149790, 617842147959019, 10821864145358779, 200955801421862020, 3943205940005194330, 81506338541922078355, 1769606318933022398611
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 145*x^4 + 1081*x^5 + 8863*x^6 +...
where, by definition,
A(x) = 1 + x*A(x)/(1+x*A(x)) + 2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +....
also, g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + 3*x^2*A(x)^2 + 12*x^3*A(x)^3 + 60*x^4*A(x)^4 + 360*x^5*A(x)^5 + 2520*x^6*A(x)^6 +...+ (n+1)!/2*x^n*A(x)^n +...
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{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m^m*x^m*A^m/(1+m*x*A+x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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{a(n)=local(B=1+sum(m=1, n, (m+1)!/2*x^m)+x*O(x^n)); polcoeff(1/x*serreverse(x/B), n)}
for(n=0, 30, print1(a(n), ", "))
A224786
G.f. satisfies: A(x) = Sum_{n>=0} x^n / (A(x) - n*x)^n.
Original entry on oeis.org
1, 1, 1, 2, 6, 23, 110, 607, 3742, 25324, 185566, 1457998, 12195992, 108010446, 1008224881, 9883048933, 101418491070, 1086613660608, 12126900841444, 140682966122152, 1693340044490513, 21111988598271746, 272228110567491910, 3625334790162237116
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 110*x^6 + 607*x^7 +...
where, by definition,
A(x) = 1 + x/(A(x) - x) + x^2/(A(x) - 2*x)^2 + x^3/(A(x) - 3*x)^3 + x^4/(A(x) - 4*x)^4 + x^5/(A(x) - 5*x)^5 +....
Also, the g.f. satisfies:
A(x) = 1 + x/A(x) + 2*x^2/A(x)^2 + 6*x^3/A(x)^3 + 23*x^4/A(x)^4 + 104*x^5/A(x)^5 + 537*x^6/A(x)^6 +...+ A080108(n)*x^n/A(x)^n +...
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{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/(A-m*x+x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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