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A249833
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E.g.f. satisfies: A(x) = 1 + Integral A(x) + A(x)^2*log(A(x)) dx.
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2
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1, 1, 2, 7, 34, 210, 1574, 13866, 140340, 1604284, 20439484, 287152488, 4409695952, 73482586464, 1320533540808, 25456195929232, 523975944225280, 11469534961767408, 266038450202037728, 6518167274358688512, 168209881653024622944, 4560447490191133853536, 129593625015740116555072
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OFFSET
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0,3
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COMMENTS
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Compare to e.g.f. of Bell numbers: if B(x) = exp(exp(x)-1) then
B(x) = 1 + Integral B(x) + B(x)*log(B(x)) dx.
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LINKS
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FORMULA
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E.g.f. satisfies: A(x) = exp( Integral 1 + A(x)*log(A(x)) dx ).
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 210*x^5/5! + ...
Related expansions.
A(x)^2*log(A(x)) = x + 5*x^2/2! + 27*x^3/3! + 176*x^4/4! + 1364*x^5/5! + ...
A(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 148*x^4/4! + 1040*x^5/5! + 8688*x^6/6! + 84068*x^7/7! + 924384*x^8/8! + 11381696*x^9/9! + ...
log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 64*x^5/5! + 424*x^6/6! + 3358*x^7/7! + 30952*x^8/8! + 325488*x^9/9! + 3845724*x^10/10! + ...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^2*log(A +x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(1+A*log(A +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) upto(n) = n--; my(v1, v2, v3, v4); v1 = vector(n + 1, i, 1); v2 = v1; v3 = vector(n + 1, i, 0); v3[1] = 1; v4 = vector(n, i, vector(i, j, (j == 1) || (j == i))); for(i = 3, n, for(j = 2, i - 1, v4[i][j] = v4[i - 1][j] + v4[i - 1][j - 1])); for(i = 1, n, for(q = 0, n - i, v2[q + 1] = v1[q + 2] + (q + 1) * sum(j = 0, q, v4[q + 1][j + 1] * v1[j + 1])); v1 = v2; v3[i + 1] = v1[1]); concat(1, v3) \\ Mikhail Kurkov, Dec 24 2023 [verification needed]
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CROSSREFS
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Column k=0 of A370382 (without initial term and with different offset).
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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