A363388
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / (k*x^k) ).
Original entry on oeis.org
1, 1, 1, 2, 5, 10, 28, 70, 190, 517, 1441, 4057, 11572, 33294, 96620, 282319, 830178, 2454384, 7292106, 21759413, 65185967, 195976025, 591097127, 1788122219, 5423917828, 16493458475, 50270190728, 153544874713, 469916030995, 1440807810639, 4425266768759, 13613578089594, 41943137192265
Offset: 1
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nmax = 33; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
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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, (-1)^k*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
A363465
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^3 / (k*x^(2*k)) ).
Original entry on oeis.org
1, 1, 1, 4, 10, 35, 113, 405, 1447, 5369, 20143, 76908, 296800, 1157784, 4554142, 18050308, 72003513, 288880549, 1164867528, 4718481975, 19190711729, 78338352168, 320851617424, 1318115448886, 5430133003281, 22427330328214, 92847100210382, 385217596191075, 1601483701650310
Offset: 1
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nmax = 29; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[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[d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]
A363466
G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^4 / (k*x^(3*k)) ).
Original entry on oeis.org
1, 1, 1, 5, 15, 61, 240, 1019, 4387, 19462, 87649, 401077, 1856698, 8685295, 40978465, 194806667, 932141498, 4486014160, 21699575863, 105443142514, 514469464550, 2519437043753, 12379461876092, 61013509071216, 301553269618318, 1494229881209940, 7421627743464582, 36942997716584746
Offset: 1
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nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[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[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.