A349505
E.g.f. satisfies: A(x) = (1 + x)^(A(x)^3).
Original entry on oeis.org
1, 1, 6, 81, 1668, 46740, 1659102, 71386602, 3611376360, 210083758704, 13817649943440, 1013979735381888, 82134894774767832, 7279520816638839600, 700730732176038359208, 72803537907677356262760, 8120227815636769492383168
Offset: 0
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nmax = 20; CoefficientList[Series[(-LambertW[-3*Log[1 + x]]/(3*Log[1 + x]))^(1/3), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)
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a(n) = sum(k=0, n, (3*k+1)^(k-1)*stirling(n, k, 1));
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N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*log(1+x)^k/k!)))
A264407
E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2).
Original entry on oeis.org
1, 1, 6, 66, 1084, 23920, 665388, 22374884, 883177328, 40043323728, 2051202965280, 117166763184768, 7384596609153696, 509084508866799840, 38108295339435463296, 3078340850588419228800, 266906341797637061659392, 24724454378396015985551616, 2436960508983873399401081856, 254658073346711773211982974976, 28122779871625104764662272952320
Offset: 0
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1084*x^4/4! + 23920*x^5/5! + 665388*x^6/6! + 22374884*x^7/7! + 883177328*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^2).
From a LambertW identity,
A(x) = 1 - log(1-x) + 5*log(1-x)^2/2! - 7^2*log(1-x)^3/3! + 9^3*log(1-x)^4/4! - 11^4*log(1-x)^5/5! + 13^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^2 + x^2*A(x)^2*(A(x)^2+1)/2! + x^3*A(x)^2*(A(x)^2+1)*(A(x)^2+2)/3! + x^4*A(x)^2*(A(x)^2+1)*(A(x)^2+2)*(A(x)^2+3)/4! +...
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Table[Sum[Abs[StirlingS1[n, k]] * (2*k+1)^(k-1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 18 2015 *)
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{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^2 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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a(n) = sum(k=0,n,abs(stirling(n, k, 1))*(2*k+1)^(k-1));
for(n=0,20,print1(a(n),", "))
A357334
E.g.f. satisfies A(x) = -log(1 - x) * exp(3 * A(x)).
Original entry on oeis.org
0, 1, 7, 101, 2286, 71064, 2815812, 135719352, 7708432680, 504204903504, 37327594368240, 3085620116373288, 281715917686701264, 28154794766366676888, 3057177180272007475368, 358397769923628731936280, 45115415964514707860498688, 6069465245766845367272738304
Offset: 0
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nmax = 17; A[_] = 0;
Do[A[x_] = -Log[1 - x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(3*log(1-x))/3)))
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a(n) = sum(k=1, n, (3*k)^(k-1)*abs(stirling(n, k, 1)));
A367200
E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(A(x)^3).
Original entry on oeis.org
1, 1, 10, 201, 6216, 261300, 13923762, 899679144, 68368215528, 5975493315264, 590584798045440, 65132133511182408, 7929604511136220536, 1056369806709005923992, 152854816585283562807288, 23873492748049623945947160, 4002988479012229417182249408
Offset: 0
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a(n) = sum(k=0, n, (n+3*k+1)^(k-1)*abs(stirling(n, k, 1)));
Showing 1-4 of 4 results.