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A353325 G.f. A(x) satisfies: A(x) = ( A(-x) + sqrt( A(-x)^2 + 64*x/A(-x) ) )/2. 3
1, 8, 28, 64, 126, 416, 1680, 5248, 13973, 53008, 224092, 768128, 2298362, 9091712, 38646080, 138919680, 445239890, 1791759736, 7578341516, 28075757440, 94326701336, 382678126688, 1606044597904, 6077887574016, 21141924364377, 86127882138832, 358591678307792 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Equals the self-convolution 8th power of A353324.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = ( A(-x) + sqrt( A(-x)^2 + 64*x/A(-x) ) )/2.
(2) A(-x) = ( A(x) + sqrt( A(x)^2 - 64*x/A(x) ) )/2.
(3) A(x)*A(-x) * (A(x) - A(-x)) = 16*x.
(4) A(-x) = (2 - A(x)^(1/8))^8.
(5) A(x) = (1 + (A(x)^(1/8) - A(-x)^(1/8))/2)^8.
(6) A(x) = ( 1 + Series_Reversion( x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) ) )^8.
EXAMPLE
G.f.: A(x) = 1 + 8*x + 28*x^2 + 64*x^3 + 126*x^4 + 416*x^5 + 1680*x^6 + 5248*x^7 + 13973*x^8 + 53008*x^9 + 224092*x^10 + ...
such that A(-x) = ( A(x) + sqrt( A(x)^2 - 64*x/A(x) ) )/2.
Related series.
A(x)^(1/8) = 1 + x + x^3 + 24*x^5 + 95*x^7 + 2699*x^9 + 15803*x^11 + 426524*x^13 + 3152930*x^15 + 78893000*x^17 + ... + A353324(n)*x^n + ...
sqrt( A(x)^2 - 64*x/A(x) ) = 1 - 24*x + 28*x^2 - 192*x^3 + 126*x^4 - 1248*x^5 + 1680*x^6 - 15744*x^7 + 13973*x^8 - 159024*x^9 + ...
A(x)*A(-x) = 1 - 8*x^2 + 12*x^4 - 336*x^6 + 686*x^8 - 38896*x^10 + 48064*x^12 - 6011040*x^14 + 132853*x^16 - 1072398368*x^18 + ...
PROG
(PARI) {a(n) = my(A=1+8*x+28*x^2, B=1-8*x+28*x^2); for(i=0, n,
A = (B + sqrt(B^2 + 64*x/B +x*O(x^n)))/2;
B = (A + sqrt(A^2 - 64*x/A +x*O(x^n)))/2;
B = (1 - (A^(1/8) - B^(1/8))/2)^8; );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Series Reversion Formula */
{a(n) = my(A = (1 + serreverse(x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) +x^2*O(x^n) ))^8); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Sequence in context: A007331 A002408 A340964 * A101127 A007259 A134747
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 13 2022
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)