A193379 Norm of coefficients in the series C(x) defined by C(x) = 1 + x*C(i*x)^2, where i^2 = -1.

1, 1, 4, 17, 20, 212, 464, 4361, 17812, 60532, 123088, 4117252, 29724752, 84585040, 430795584, 8219554697, 47479991380, 214977407060, 898098431312, 16268050731620, 98128441675472, 417822285118032, 1654860158000960, 35730391312348996, 243329575991962320

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..601

Formula

a(n) = A193377(n)^2 + A193378(n)^2. - Paul D. Hanna, Jan 20 2026

Example

G.f. A(x) = 1 + x + 4*x^2 + 17*x^3 + 20*x^4 + 212*x^5 + 464*x^6 + 4361*x^7 + 17812*x^8 + 60532*x^9 + 123088*x^10 + ...
A related series is defined by C(x) = 1 + x*C(i*x)^2, where
C(x) = 1 + x + 2*i*x^2 + (-1 - 4*i)*x^3 + (-4 + 2*i)*x^4 + (-14 - 4*i)*x^5 + (8 - 20*i)*x^6 + (35 + 56*i)*x^7 + (44 - 126*i)*x^8 + (246 - 4*i)*x^9 + (168 + 308*i)*x^10 + ...
and the norm of the complex coefficients of C(x) generates this sequence starting with a(0) = 1; a(1) = 1;
  a(2) = 2^2 = 4;
  a(3) = (-1)^2 + (-4)^2 = 17;
  a(4) = (-4)^2 + 2^2 = 20;
  a(5) = (-14)^2 + (-4)^2 = 212;
  ...
The square of C(x) begins
C(x)^2 = 1 + 2*x + (1 + 4*i)*x^2 + (-2 - 4*i)*x^3 + (-14 - 4*i)*x^4 + (-20 - 8*i)*x^5 + (-35 - 56*i)*x^6 + (126 + 44*i)*x^7 + (246 - 4*i)*x^8 + ...
The real part of C(x) begins
real(C(x)) = 1 + x - x^3 - 4*x^4 - 14*x^5 + 8*x^6 + 35*x^7 + 44*x^8 + 246*x^9 + 168*x^10 - 1906*x^11 + 296*x^12 + ... + A193377(n)*x^n + ...
The imaginary part of C(x) begins
imag(C(x)) = 2*x^2 - 4*x^3 + 2*x^4 - 4*x^5 - 20*x^6 + 56*x^7 - 126*x^8 - 4*x^9 + 308*x^10 - 696*x^11 + 5444*x^12 + ... + A193378(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + x*subst(A, x, I*x + x*O(x^n))^2); norm(polcoef(A, n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A193377 (real), A193378 (imag).

Cf. A138157.

Sequence in context: A128981 A212748 A032828 * A022134 A041529 A042033

Adjacent sequences: A193376 A193377 A193378 * A193380 A193381 A193382

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2011

Extensions

Entry revised by Paul D. Hanna, Jan 20 2026

Status

approved