A204190 E.g.f. A(x) satisfies: Laplace(A(x)) = Sum_{n>=0} Laplace(A(x)^n) * x^n.

1, 1, 2, 5, 15, 54, 231, 1151, 6544, 41759, 295373, 2293276, 19387721, 177278761, 1743221942, 18342831993, 205649438307, 2447327139282, 30810764360107, 409126482916771, 5714631215634236, 83760248865355379, 1285427577602987313, 20613047005064025320

Offset

0,3

Comments

Here Laplace( Sum_{n>=0} c(n)*x^n/n! ) = Sum_{n>=0} c(n)*x^n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..155 (terms 0..100 from Paul D. Hanna)

Formula

E.g.f. satisfies: A(x) = Sum_{n>=0} Integral^n A(x)^n dx^n, where integral^n A(x)^n dx^n denotes the n-th integration of A(x)^n with no constant of integration. - Paul D. Hanna, Dec 29 2013

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 54*x^5/5! +...
The table of coefficients in the Laplace transform of A(x)^n begins:
n=0: [1, 0,  0,   0,    0,     0,       0,        0,         0, ...];
n=1: [1, 1,  2,   5,   15,    54,     231,     1151,      6544, ...];
n=2: [1, 2,  6,  22,   94,   458,    2510,    15322,    103366, ...];
n=3: [1, 3, 12,  57,  309,  1872,   12507,    91317,    723738, ...];
n=4: [1, 4, 20, 116,  756,  5436,   42612,   361100,   3287428, ...];
n=5: [1, 5, 30, 205, 1555, 12890,  115535,  1111195,  11401900, ...];
n=6: [1, 6, 42, 330, 2850, 26694,  268626,  2884326,  32869914, ...];
n=7: [1, 7, 56, 497, 4809, 50148,  558915,  6616337,  82785766, ...];
n=8: [1, 8, 72, 712, 7624, 87512, 1068872, 13813432, 188042728, ...];
n=9: [1, 9, 90, 981,11511,144126, 1912887, 26774775, 393614568, ...]; ...
where the antidiagonal sums equal this sequence:
1 + 1 + 0 = 2;
1 + 2 + 2 + 0 = 5;
1 + 3 + 6 + 5 + 0 = 15;
1 + 4 + 12 + 22 + 15 + 0 = 54; ...

Prog

(PARI) a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, serlaplace(serconvol(A+x*O(x^n), exp(x+x*O(x^n)))^m+x*O(x^n))*x^m)+x*O(x^n)); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From e.g.f. involving iterated integration: */
{INTEGRATE(n, F)=local(G=F); for(i=1, n, G=intformal(G)); G}
{a(n)=local(A=1+x); for(i=1, n, A=1+sum(k=1, n, INTEGRATE(k, (A+x*O(x^n))^k))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 29 2013

Crossrefs

Cf. A213058.

Sequence in context: A107112 A193318 A171450 * A051295 A334154 A009383

Adjacent sequences: A204187 A204188 A204189 * A204191 A204192 A204193

Keyword

nonn

Author

Paul D. Hanna, Jun 07 2012

Status

approved