A367719 -id:A367719 - cp's OEIS frontend

A143567 E.g.f. satisfies A(x) = exp(x*A(x^3/3!)).

1, 1, 1, 1, 5, 21, 61, 211, 1401, 8065, 37241, 240021, 1997821, 14657501, 105629525, 958412911, 9201199281, 86311594881, 871038486001, 9432024424585, 106531641929781, 1271523772132741, 15583607760968941, 194983864950339851

Offset: 0

Alois P. Heinz, Aug 24 2008, Aug 25 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

3rd column of A143565.

Cf. A367719.

A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-3)(x^3/6)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..29);

A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^3/3!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient [A[n][x], x, n]*n!; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (3*k+1) * a(k) * a(n-1-3*k) / (6^k * k! * (n-1-3*k)!). - Seiichi Manyama, Nov 28 2023

A367720 E.g.f. satisfies A(x) = exp(x*A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 196561, 3659041, 29993041, 159762241, 1686639241, 60298558321, 987112886761, 9315623640961, 76611297104161, 2454331471018561, 69805324167893281, 1086439146068753281, 62621251106366355481, 1358219171406244427281

Offset: 0

Seiichi Manyama, Nov 28 2023

nonn

This sequence is different from A354554.

Cf. A138292, A367719.

Cf. A354554.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\4, (4*j+1)*v[j+1]*v[i-4*j]/(j!*(i-1-4*j)!))); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (4*k+1) * a(k) * a(n-1-4*k) / (k! * (n-1-4*k)!).

A367748 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2) * A(x^3)).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 40153, 434673, 5231593, 71641891, 1051099941, 16673401273, 289579557853, 5336892479283, 104679769486321, 2199654959898721, 48682858073000913, 1135029742772833603, 27968526823231736893, 721419709386600131241

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A367747, A367749, A367751.

Cf. A367652, A367719.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\3+1]*v[i-j]/((j\3)!*(i-1-j)!))); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/3)) * a(n-1-k) / (floor(k/3)! * (n-1-k)!).

cp's OEIS Frontend

A143567 E.g.f. satisfies A(x) = exp(x*A(x^3/3!)).

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A367720 E.g.f. satisfies A(x) = exp(x*A(x^4)).

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A367748 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2) * A(x^3)).

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