A268170 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero.
1, 1, 2, 5, 16, 65, 326, 1947, 13410, 104181, 900214, 8566655, 89055224, 1004141647, 12204369138, 159036267519, 2211764983734, 32696763676521, 511987792322430, 8465194670035767, 147370831072230860, 2694506417687396995, 51622643862824956898, 1034153511794063402519, 21621325640846679627146
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +... such that log(A(x)) = Integral B(x) dx where B(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...+ A268171(n)*x^n/n! +... and A(x) and B(x) satisfy: (1) A(x) = B'(x)/B(x) + exp(x) - 1, (2) B(x) = A'(x)/A(x), (3) log(A(x)) = Integral B(x) dx, (4) log(B(x)) = Integral A(x) dx + 1+x - exp(x). RELATED SERIES. log(A(x)) = x + x^2/2! + x^3/3! + 2*x^4/4! + 9*x^5/5! + 46*x^6/6! + 245*x^7/7! + 1474*x^8/8! + 10315*x^9/9! + 82174*x^10/10! + 726591*x^11/11! + 7038632*x^12/12! +... Let J(x) equal the series reversion of log(A(x)); then J(x) = x - x^2/2! + 2*x^3/3! - 7*x^4/4! + 31*x^5/5! - 172*x^6/6! + 1155*x^7/7! - 9027*x^8/8! + 80676*x^9/9! - 811727*x^10/10! + 9075333*x^11/11! - 111633356*x^12/12! +... where A(J(x)) = exp(x).
Programs
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PARI
{a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp(1+x - exp(x +x*O(x^n)) + intformal( A ) ) ); n!*polcoeff(A, n)} for(n=0, 30, print1(a(n), ", "))
Comments