A363387
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).
Original entry on oeis.org
1, 1, 1, 3, 6, 17, 42, 120, 330, 962, 2797, 8334, 24989, 75905, 232142, 715830, 2220473, 6928411, 21723883, 68424327, 216376757, 686742855, 2186771571, 6984248840, 22368127861, 71818903891, 231132440916, 745454242656, 2409080380316, 7799945417349
Offset: 1
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nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]
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seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(sum(k=1, m, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023
A363467
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 / (k*x^(2*k)) ).
Original entry on oeis.org
1, 1, 1, 3, 9, 25, 88, 292, 1031, 3685, 13433, 49608, 185465, 699963, 2664650, 10217130, 39428179, 153009240, 596761737, 2337875430, 9195732624, 36301739221, 143780858517, 571191310205, 2275409450019, 9087376470138, 36377539265376, 145937953205705, 586645566919856
Offset: 1
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nmax = 29; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^3/(k x^(2 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]
A363468
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^4 / (k*x^(3*k)) ).
Original entry on oeis.org
1, 1, 1, 4, 14, 48, 201, 812, 3455, 14961, 65954, 294884, 1334526, 6098879, 28114885, 130561444, 610244889, 2868547475, 13552299256, 64316483918, 306473091394, 1465727378317, 7033293786125, 33851816310445, 163384902125185, 790589562321385, 3834540111072545, 18638976010097900
Offset: 1
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nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^4/(k x^(3 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[f[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1)d g[d + 3], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]
Showing 1-3 of 3 results.