A218673 -id:A218673

Search: a218673 -id:a218673

Displaying 1-4 of 4 results found.

page 1

A218672

O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

+10
26

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare to the LambertW identities:

(1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..50

EXAMPLE

O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...

where

A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...

simplifies to a power series in x with integer coefficients.

MATHEMATICA

a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];

a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A, x, k*x)^k/k!*exp(-k*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A218673, A218674, A218675, A218676, A218681.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

Cf. A193363, A221409, A221410, A221411, A221412, A221413.

Cf. A209276, A209277.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 04 2012

STATUS

approved

A218674

O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).

+10
4

1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

LINKS

Table of n, a(n) for n=0..17.

EXAMPLE

O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...

where

A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...

simplifies to a power series in x with integer coefficients.

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A^3, x, k*x)^k/k!*exp(-k*x*subst(A^3, x, k*x)+x*O(x^n)))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A218672, A218673, A218675, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 04 2012

STATUS

approved

A218675

O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).

+10
4

1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

LINKS

Table of n, a(n) for n=0..16.

EXAMPLE

O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...

where

A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...

simplifies to a power series in x with integer coefficients.

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A^4, x, k*x)^k/k!*exp(-k*x*subst(A^4, x, k*x)+x*O(x^n)))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A218672, A218673, A218674, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 04 2012

STATUS

approved

A218676

O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).

+10
4

1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

LINKS

Table of n, a(n) for n=0..16.

EXAMPLE

O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...

where

A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...

simplifies to a power series in x with integer coefficients.

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^k*x^k*subst(A^5, x, k*x)^k/k!*exp(-k*x*subst(A^5, x, k*x)+x*O(x^n)))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A218672, A218673, A218674, A218675.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 04 2012

STATUS

approved

page 1

Search completed in 0.006 seconds