A230317
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} A(k*x).
Original entry on oeis.org
1, 1, 3, 19, 215, 4016, 119092, 5503205, 393154477, 43298176708, 7340865004766, 1913028475033699, 764596866776205619, 467416620805272150858, 435773832975324764799712, 617753035395626539657324801, 1327738980050724547857227605753, 4314831625390935798178255342966024
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 215*x^4/4! + 4016*x^5/5! +...
where
A(x) = 1 + x*A(x) + x^2*A(x)*A(2*x)/2! + x^3*A(x)*A(2*x)*A(3*x)/3! + x^4*A(x)*A(2*x)*A(3*x)*A(4*x)/4! + x^5*A(x)*A(2*x)*A(3*x)*A(4*x)*A(5*x)/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))))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
A230320
E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} ( LambertW(-k*x)/(-k*x) )^(1/k).
Original entry on oeis.org
1, 1, 3, 16, 137, 1746, 31627, 785149, 25715377, 1070214364, 54862242971, 3385895548839, 247409460018217, 21118696317592498, 2080845352648353215, 234093630772343822281, 29777361783749418754593, 4247066958924682143019576, 674393753569770072828136819
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)^(1/k) = Sum_{j>=0} (k*j+1)^(j-1)*x^j/j!
where
A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)^(1/2)/2! + x^3*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)/3! + x^4*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)*W(4*x)^(1/4)/4! +...
RELATED EXPANSIONS:
W(1*x)^(1/1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
W(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + 14641*x^5/5! +...
W(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + 65536*x^5/5! +...
W(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + 194481*x^5/5! +...
W(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + 456976*x^5/5! +...
W(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 15625*x^4/4! + 923521*x^5/5! +...
W(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 24389*x^4/4! + 1679616*x^5/5! +...
W(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 35937*x^4/4! + 2825761*x^5/5! +...
W(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 50653*x^4/4! + 4477456*x^5/5! +...
W(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 68921*x^4/4! + 6765201*x^5/5! +...
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{a(n)=local(LambertW=serreverse(x*exp(x+x^2*O(x^n))), A=1);
A=sum(m=0, n, x^m/m!*prod(k=1, m, (subst(LambertW, x, -k*x)/(-k*x))^(1/k)));
n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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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)^(1/k)));
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+1)^(j-1)*x^j/j!)+x*O(x^n) ));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A230323
E.g.f.: Sum_{n>=0} x^n/n! * LambertW(-n*x)/(-n*x).
Original entry on oeis.org
1, 1, 3, 16, 149, 2196, 47167, 1380394, 52206345, 2457554248, 140408870651, 9563233300974, 765251809488157, 71012512562009500, 7552175961721086711, 911014865916673379026, 123562591612443767093393, 18704896422725902820936976, 3140424504257773679216307955
Offset: 0
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2196*x^5/5! +...
Let W(x) = LambertW(-x)/(-x), then
W(n*x) = Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k! and
W(n*x) = [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n
where
A(x) = 1 + x*W(x) + x^2*W(2*x)/2! + x^3*W(3*x)/3! + x^4*W(4*x)/4! + x^5*W(5*x)/5! + x^6*W(6*x)/6! +...
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! +...
...
W(1*x) = (1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...)^1
W(2*x) = (1 + x + 5*x^2/2! + 7^2*x^3/3! + 9^3*x^4/4! + 11^4*x^5/5! +...)^2
W(3*x) = (1 + x + 7*x^2/2! + 10^2*x^3/3! + 13^3*x^4/4! + 16^4*x^5/5! +...)^3
W(4*x) = (1 + x + 9*x^2/2! + 13^2*x^3/3! + 17^3*x^4/4! + 21^4*x^5/5! +...)^4
W(5*x) = (1 + x + 11*x^2/2! + 16^2*x^3/3! + 21^3*x^4/4! + 26^4*x^5/5! +...)^5
...
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Flatten[{1,Table[Sum[Binomial[n,k] * (n-k)^k * (k+1)^(k-1),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 29 2014 *)
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{a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k*(k+1)^(k-1))}
for(n=0,20,print1(a(n),", "))
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{a(n)=local(A=1);
A=sum(m=0,n,x^m/m!*sum(j=0,n,m^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(A=1);
A=sum(m=0,n,x^m/m!*sum(j=0,n,(m*j+1)^(j-1)*x^j/j! +x*O(x^n))^m );
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
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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!*subst(W,x,m*x));
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=1+sum(m=1,n,x^m/m!*subst(LambertW,x,-m*x)/(-m*x));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
Showing 1-3 of 3 results.