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A291387
p-INVERT of (1,1,0,0,0,0,...), where p(S) = (1 - 4 S)^2.
3
8, 56, 352, 2096, 12032, 67328, 369664, 2000128, 10696704, 56666112, 297836544, 1555066880, 8073379840, 41709076480, 214558048256, 1099562549248, 5616171483136, 28599668703232, 145249047412736, 735884541427712, 3720035809886208, 18767645931208704
OFFSET
0,1
COMMENTS
Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
See A291382 for a guide to related sequences.
All terms = 0 mod 8. - Muniru A Asiru, Sep 07 2017
FORMULA
G.f.: -((8 (1 + x) (-1 + 2 x + 2 x^2))/(-1 + 4 x + 4 x^2)^2).
a(n) = 8*a(n-1) - 8*a(n-2) - 32*a(n-3) - 16*a(n-4) for n >= 5.
MATHEMATICA
z = 60; s = x + x^2; p = (1 - 4 s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A019590 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291387 *)
u / 8 (* A291388 *)
PROG
(GAP)
a:=8*[1, 7, 44, 262];; for n in [5..10^2] do a[n]:=8*a[n-1]-8*a[n-2]-32*a[n-3]-16*a[n-4]; od; a; # Muniru A Asiru, Sep 07 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 04 2017
STATUS
approved