A161566 -id:A161566 - cp's OEIS frontend

A161565 E.g.f. satisfies: A(x) = exp(xexp(2x*A(x))).

1, 1, 5, 37, 417, 6201, 115393, 2583141, 67643201, 2029868785, 68699859201, 2589393498429, 107580709769569, 4885086832499433, 240716442970201409, 12793428673619226901, 729511897042502788737, 44427614415877495608801

Offset: 0

Paul D. Hanna, Jun 14 2009

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! +...
log(A(x)) = x*B(x) where B(x) = exp(2*x*A(x)) = e.g.f. of A161566:
B(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..367

Cf. A161566.

Flatten[{1,Table[Sum[2^(n-k) * Binomial[n,k] * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)

{a(n)=sum(k=0,n,2^(n-k)*binomial(n,k) * (n-k+1)^(k-1) * k^(n-k))}

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*exp(2*x*A+O(x^n))));n!*polcoeff(A,n,x)}

a(n) = Sum_{k=0..n} 2^(n-k) * C(n,k) * (n-k+1)^(k-1) * k^(n-k).

a(n) ~ sqrt(LambertW(1/r)) * n^(n-1) / (2*exp(n)*r^(n+1)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014

A161567 E.g.f. satisfies: A(x) = exp(xexp(xA(x)^2)).

Original entry on oeis.org

1, 1, 3, 22, 233, 3356, 61057, 1343686, 34731377, 1031493880, 34617603041, 1295705404874, 53516386593001, 2417918198462404, 118628419305036929, 6280926119941402486, 356960234149564116833, 21674784895404653181680

Offset: 0

Paul D. Hanna, Jun 14 2009

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! +...
log(A(x)) = x*G(x) where G(x) = exp(x*A(x)^2) = e.g.f. of A161565:
G(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! + 6201*x^5/5! +...
A(x)^2 = e.g.f. of A161566:
A(x)^2 = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..365

Cf. A161565, A161566.

Flatten[{1,Table[Sum[Binomial[n,k] * (2*(n-k) + 1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 28 2014 *)

{a(n)=sum(k=0,n,binomial(n,k)*(2*(n-k)+1)^(k-1)*k^(n-k))}

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*exp(x*A^2+O(x^n))));n!*polcoeff(A,n,x)}

a(n) = Sum_{k=0..n} C(n,k) * (2*(n-k) + 1)^(k-1) * k^(n-k).
E.g.f.: A(x) = F(x)^(1/2) where F(x) = e.g.f. of A161566.
E.g.f.: A(x) = exp(x*G(x)) where G(x) = e.g.f. of A161565.
a(n) ~ n^(n-1) / (2*exp(n)*r^(n+1/2)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014

A161552 E.g.f. satisfies: A(x,y) = exp(xyexp(x*A(x,y))).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 12, 1, 0, 4, 72, 48, 1, 0, 5, 320, 810, 160, 1, 0, 6, 1200, 8640, 6480, 480, 1, 0, 7, 4032, 70875, 143360, 42525, 1344, 1, 0, 8, 12544, 489888, 2240000, 1792000, 244944, 3584, 1, 0, 9, 36864, 3000564, 27869184, 49218750, 18579456, 1285956, 9216, 1

Offset: 0

Paul D. Hanna, Jun 13 2009, Jun 14 2009

E.g.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n!.

Row sums, (n+1)^(n-1), equal A000272 (number of trees on n labeled nodes).

			Triangle begins:
1;
0,1;
0,2,1;
0,3,12,1;
0,4,72,48,1;
0,5,320,810,160,1;
0,6,1200,8640,6480,480,1;
0,7,4032,70875,143360,42525,1344,1;
0,8,12544,489888,2240000,1792000,244944,3584,1;
0,9,36864,3000564,27869184,49218750,18579456,1285956,9216,1; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000272, A072590; A161565, A161566, A161567, A141369.

Join[{1}, Table[Binomial[n, k]*(n - k + 1)^(k - 1)*k^(n - k), {n, 1, 10}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 18 2017 *)

{T(n,k)=binomial(n,k)*(n-k+1)^(k-1)*k^(n-k)}

{T(n,k)=local(A=1+x); for(i=0,n, A=exp(x*y*exp(x*A+O(x^n)))); n!*polcoeff(polcoeff(A,n,x),k,y)}

T(n,k) = binomial(n,k) * (n-k+1)^(k-1) * k^(n-k).
E.g.f. A(x,y) at y=1: A(x,1) = LambertW(-x)/(-x).
From Paul D. Hanna, Jun 14 2009: (Start)
More generally, if G(x) = exp(p*x*exp(q*x*G(x))),
where G(x)^m = Sum_{n>=0} g(n,m)*x^n/n!,
then g(n,m) = Sum_{k=0..n} C(n,k)*p^k*q^(n-k)*m*(n-k+m)^(k-1)*k^(n-k).
(End)

A161568 E.g.f. satisfies: A(x) = exp(2xexp(3xA(x))).

Original entry on oeis.org

1, 2, 16, 206, 3976, 101402, 3237220, 124293206, 5582747824, 287346080690, 16680250440124, 1078327289938670, 76840445565238024, 5984507179839282122, 505778795448930860308, 46104043794638089809158

Offset: 0

Paul D. Hanna, Jun 14 2009

nonn

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 206*x^3/3! + 3976*x^4/4! +...

G. C. Greubel, Table of n, a(n) for n = 0..341

Cf. A161565, A161566, A161567.

Flatten[{1,Table[Sum[Binomial[n,k] * 2^k * 3^(n-k) * (n-k+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 15 2014 *)

{a(n)=sum(k=0,n,binomial(n,k)*2^k*3^(n-k)*(n-k+1)^(k-1)*k^(n-k))}

{a(n)=local(A=1+x);for(i=0,n,A=exp(2*x*exp(3*x*A+O(x^n))));n!*polcoeff(A,n,x)}

a(n) = Sum_{k=0..n} C(n,k) * 2^k * 3^(n-k) * (n-k+1)^(k-1) * k^(n-k).
More generally, if G(x) = exp(p*x*exp(q*x*G(x))) = Sum_{n>=0} g(n)*x^n/n!,
then g(n) = Sum_{k=0..n} C(n,k) * p^k * q^(n-k) * (n-k+1)^(k-1) * k^(n-k).
a(n) ~ sqrt(s/3) * n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.149417197143691584817... and s = 2.468671804906329807069... are roots of the system of equations 3*r*s*Log(s) = 1, 6*exp(3*r*s)*s*r^2 = 1. - Vaclav Kotesovec, Jul 15 2014

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A161565 E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))).

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A161567 E.g.f. satisfies: A(x) = exp(x*exp(x*A(x)^2)).

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A161552 E.g.f. satisfies: A(x,y) = exp(x*y*exp(x*A(x,y))).

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A161568 E.g.f. satisfies: A(x) = exp(2*x*exp(3*x*A(x))).

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A161565 E.g.f. satisfies: A(x) = exp(xexp(2x*A(x))).

A161567 E.g.f. satisfies: A(x) = exp(xexp(xA(x)^2)).

A161552 E.g.f. satisfies: A(x,y) = exp(xyexp(x*A(x,y))).

A161568 E.g.f. satisfies: A(x) = exp(2xexp(3xA(x))).