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A301309 G.f.: Sum_{n>=0} ( (1+x)^n + (1-x)^n )^n / 2^(2*n+1), an even function. 2
1, 5, 418, 97248, 44494788, 33701146040, 38158722166012, 60370440881763184, 127193089522406873576, 344265367844128036044688, 1164086577885251318385747568, 4808913945776510766505317067088, 23831677319262549731059823149874928, 139543211306816620890086979219586374480, 953076439362156646686630002626476525309552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Is there a finite expression for the terms of this sequence?

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

G.f.: Sum_{n>=0} [ Sum_{k=0..[n/2]} binomial(n,2*k) * x^(2*k) ]^n / 2^(n+1).

a(n) ~ c * d^n * n!^2 / n, where d = 37.4848548470528901759474480740698513182712... and c = 0.1647617452257182061114277957479516654825... - Vaclav Kotesovec, Oct 07 2020

EXAMPLE

G.f.: A(x) = 1 + 5*x^2 + 418*x^4 + 97248*x^6 + 44494788*x^8 + 33701146040*x^10 + 38158722166012*x^12 + 60370440881763184*x^14 + ...

such that

A(x) = 1/2  +  ((1+x) + (1-x))/2^3  +  ((1+x)^2 + (1-x)^2)^2/2^5  +  ((1+x)^3 + (1-x)^3)^3/2^7  +  ((1+x)^4 + (1-x)^4)^4/2^9  +  ((1+x)^5 + (1-x)^5)^5/2^11 + ...

Equivalently,

A(x) = 1/2 + 1/2^2 + (1 + x^2)^2/2^3 + (1 + 3*x^2)^3/2^4 + (1 + 6*x^2 + x^4)^4/2^5 + (1 + 10*x^2 + 5*x^4)^5/2^6 + (1 + 15*x^2 + 15*x^4 + x^6)^6/2^7 + ...

CROSSREFS

Cf. A301308, A302108.

Sequence in context: A221700 A216089 A201887 * A218406 A217939 A147684

Adjacent sequences:  A301306 A301307 A301308 * A301310 A301311 A301312

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 18 2018

STATUS

approved

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Last modified May 27 21:14 EDT 2022. Contains 354110 sequences. (Running on oeis4.)