A230321
E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} LambertW(-k*x)/(-k*x).
Original entry on oeis.org
1, 1, 3, 19, 203, 3296, 75862, 2340710, 92647529, 4554162028, 271560907586, 19291412245084, 1608220567355569, 155269058419296810, 17162375811312467478, 2150431928188151079196, 302882085165757326494593, 47608431893360236873620584, 8298673711523249659301551906
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1746*x^5/5! +...
Let W(x) = LambertW(-x)/(-x), then
W(k*x) = Sum_{j>=0} k^j*(j+1)^(j-1)*x^j/j!
where
A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)/2! + x^3*W(x)*W(2*x)*W(3*x)/3! + x^4*W(x)*W(2*x)*W(3*x)*W(4*x)/4! + x^5*W(x)*W(2*x)*W(3*x)*W(4*x)*W(5*x)/5! +...
RELATED EXPANSIONS:
W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...
W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...
W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...
W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...
...
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{a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n),A=1);
A=sum(m=0,n,x^m/m!*prod(k=1,m,subst(W,x,k*x)));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
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{a(n)=local(A=1);
A=sum(m=0,n,x^m/m!*prod(k=1,m,sum(j=0,n,k^j*(j+1)^(j-1)*x^j/j!)+x*O(x^n) ));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
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{a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);
A=sum(m=0,n,1/m!^2*prod(k=1,m,subst(-LambertW,x,-k*x)));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A178087
G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} A(k*x).
Original entry on oeis.org
1, 1, 2, 6, 25, 141, 1071, 11011, 154739, 3005187, 81434048, 3101253384, 166823865867, 12719913809811, 1378095292930494, 212524751143894194, 46713381928627546015, 14648866052370410611923, 6558913185973371123604314, 4195585528812861561212654010
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 141*x^5 +...
where
A(x) = 1 + x*A(x) + x^2*A(x)*A(2*x) + x^3*A(x)*A(2*x)*A(3*x) + x^4*A(x)*A(2*x)*A(3*x)*A(4*x) + x^5*A(x)*A(2*x)*A(3*x)*A(4*x)*A(5*x) +...
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{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*prod(k=1, m, subst(A, x, k*x+x*O(x^n))))); polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
A230318
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} A(k*x)^(1/k).
Original entry on oeis.org
1, 1, 3, 16, 137, 1806, 36307, 1120099, 53543737, 3987172576, 462225323291, 83073483132399, 23029319036705797, 9800155391415000808, 6374883018317726171183, 6314677980863430818696221, 9492558648757349054617246833, 21588387865629652184260447567056
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1806*x^5/5! + ...
where
A(x) = 1 + x*A(x) + x^2*A(x)*A(2*x)^(1/2)/2! + x^3*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)/3! + x^4*A(x)*A(2*x)^(1/2)*A(3*x)^(1/3)*A(4*x)^(1/4)/4! + ...
RELATED EXPANSIONS.
A(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 825*x^4/4! + 22321*x^5/5! + ...
A(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2521*x^4/4! + 103606*x^5/5! + ...
A(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 5681*x^4/4! + 313521*x^5/5! + ...
A(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 10761*x^4/4! + 745726*x^5/5! + ...
A(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 18217*x^4/4! + 1519681*x^5/5! + ...
A(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 28505*x^4/4! + 2780646*x^5/5! + ...
A(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 42081*x^4/4! + 4699681*x^5/5! + ...
A(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 59401*x^4/4! + 7473646*x^5/5! + ...
A(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 80921*x^4/4! + 11325201*x^5/5! + ...
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{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m/m!*prod(k=1, m, subst(A, x, k*x+x*O(x^n))^(1/k)))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.