A213058 E.g.f. A(x) satisfies: 1-x = Sum_{n>=0} Laplace(A(x)^n) * (-x)^n.

1, 1, 1, 2, 4, 12, 38, 150, 648, 3218, 17476, 105150, 686012, 4858602, 36966912, 301473146, 2619548392, 24181633098, 236214717380, 2435055354798, 26413257186156, 300761256373298, 3586799346195376, 44710179730250874, 581430401788576664, 7874808892125746226

Offset

0,4

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..220 (terms 0..100 from Paul D. Hanna)

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 4*x^4/4! + 12*x^5/5! + 38*x^6/6! +...
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,  1,   2,    4,    12,     38,     150,      648, ...];
n=2: [1, 2,  4,  10,   30,   104,    420,    1896,     9632, ...];
n=3: [1, 3,  9,  30,  114,   486,   2316,   12210,    70632, ...];
n=4: [1, 4, 16,  68,  316,  1608,   8936,   54024,   353496, ...];
n=5: [1, 5, 25, 130,  720,  4280,  27330,  187230,  1372640, ...];
n=6: [1, 6, 36, 222, 1434,  9792,  70908,  544800,  4437288, ...];
n=7: [1, 7, 49, 350, 2590, 20034, 162680, 1389066, 12474672, ...];
n=8: [1, 8, 64, 520, 4344, 37616, 339216, 3193200, 31407632, ...];
n=9: [1, 9, 81, 738, 6876, 65988, 655326, 6752934, 72308376, ...]; ...
such that the alternating sums of antidiagonals vanish for n>1:
1 - 1 = 0;
1 - 2 + 1 = 0;
2 - 4 + 3 - 1 = 0;
4 - 10 + 9 - 4 + 1 = 0;
12 - 30 + 30 - 16 + 5 - 1 = 0;
38 - 104 + 114 - 68 + 25 - 6 + 1 = 0; ...

Prog

(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=Vec(sum(m=1, #A, serlaplace(serconvol(Ser(A), exp(x+O(x^#A)))^m)*(-x)^m))[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A204190.

Sequence in context: A149846 A341240 A108532 * A268069 A215071 A180487

Adjacent sequences: A213055 A213056 A213057 * A213059 A213060 A213061

Keyword

nonn

Author

Paul D. Hanna, Jun 06 2012

Status

approved