A204190 - OEIS

A204190

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

%I #24 Aug 12 2021 03:54:53

%S 1,1,2,5,15,54,231,1151,6544,41759,295373,2293276,19387721,177278761,

%T 1743221942,18342831993,205649438307,2447327139282,30810764360107,

%U 409126482916771,5714631215634236,83760248865355379,1285427577602987313,20613047005064025320

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

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

%H Vaclav Kotesovec, <a href="/A204190/b204190.txt">Table of n, a(n) for n = 0..155</a> (terms 0..100 from Paul D. Hanna)

%F 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

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 54*x^5/5! +...

%e The table of coefficients in the Laplace transform of A(x)^n begins:

%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=1: [1, 1, 2, 5, 15, 54, 231, 1151, 6544, ...];

%e n=2: [1, 2, 6, 22, 94, 458, 2510, 15322, 103366, ...];

%e n=3: [1, 3, 12, 57, 309, 1872, 12507, 91317, 723738, ...];

%e n=4: [1, 4, 20, 116, 756, 5436, 42612, 361100, 3287428, ...];

%e n=5: [1, 5, 30, 205, 1555, 12890, 115535, 1111195, 11401900, ...];

%e n=6: [1, 6, 42, 330, 2850, 26694, 268626, 2884326, 32869914, ...];

%e n=7: [1, 7, 56, 497, 4809, 50148, 558915, 6616337, 82785766, ...];

%e n=8: [1, 8, 72, 712, 7624, 87512, 1068872, 13813432, 188042728, ...];

%e n=9: [1, 9, 90, 981,11511,144126, 1912887, 26774775, 393614568, ...]; ...

%e where the antidiagonal sums equal this sequence:

%e 1 + 1 + 0 = 2;

%e 1 + 2 + 2 + 0 = 5;

%e 1 + 3 + 6 + 5 + 0 = 15;

%e 1 + 4 + 12 + 22 + 15 + 0 = 54; ...

%o (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)

%o for(n=0,20,print1(a(n),", "))

%o (PARI) /* From e.g.f. involving iterated integration: */

%o {INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}

%o {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)}

%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 29 2013

%Y Cf. A213058.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 07 2012