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A266908 G.f. A(x) satisfies: x = Sum_{n>=1} x^n * Product_{k=1..n} A(-k*x). 1
1, 1, 2, 7, 45, 578, 15523, 872933, 101606058, 24244705427, 11781372347197, 11604013148951290, 23086334686919094283, 92540040424223196349213, 745956027717908362991989762, 12075313247950868952015337447195, 392133966660263237084551188748738021, 25526209248562553823289966078580478182370, 3328929359036770266032093183197212572084089475, 869367378139090634989087434871564181449768656675093 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to: x = Sum_{n>=1} x^n * F(-x)^n if F(x) = 1/(1-x).

LINKS

Table of n, a(n) for n=0..19.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 45*x^4 + 578*x^5 + 15523*x^6 + 872933*x^7 + 101606058*x^8 + 24244705427*x^9 + 11781372347197*x^10 +...

where

x = x*A(-x) + x^2*A(-x)*A(-2*x) + x^3*A(-x)*A(-2*x)*A(-3*x) + x^4*A(-x)*A(-2*x)*A(-3*x)*A(-4*x) + x^5*A(-x)*A(-2*x)*A(-3*x)*A(-4*x)*A(-5*x) +...

The array of coefficients in [Product_{k=1..n} A(-k*x)] begins:

n=1: [1, -1, 2, -7, 45, -578, 15523, -872933, ...];

n=2: [1, -3, 12, -75, 851, -20052, 1030839, -113682051, ...];

n=3: [1, -6, 39, -354, 5504, -177612, 12901857, -2062308438, ...];

n=4: [1, -10, 95, -1150, 22376, -889420, 81529745, -16832853850, ...];

n=5: [1, -15, 195, -3000, 69751, -3229425, 351383145, -88131849450, ...];

n=6: [1, -21, 357, -6762, 182791, -9528099, 1183349175, -346858512000, ...];

n=7: [1, -28, 602, -13720, 423577, -24310860, 3353035806, -1117962684216, ...];

n=8: [1, -36, 954, -25704, 895065, -55714068, 8361452286, -3107960056872, ...]; ...

in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

A[#A]=(-1)^(#A)*Vec(sum(m=1, #A, prod(k=1, m, subst(Ser(A), x, -k*x))*x^m))[#A] ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

(PARI) /* Quick print of terms 0..30 */

{A=[1]; for(i=1, 30, A=concat(A, 0);

A[#A]=(-1)^(#A)*Vec(sum(n=1, #A, prod(k=1, n, subst(Ser(A), x, -k*x))*x^n))[#A] ); A}

CROSSREFS

Cf. A266812.

Sequence in context: A162045 A153549 A079491 * A162046 A162047 A162048

Adjacent sequences:  A266905 A266906 A266907 * A266909 A266910 A266911

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 05 2016

STATUS

approved

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Last modified February 20 21:05 EST 2019. Contains 320346 sequences. (Running on oeis4.)