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A319466 G.f.: Sum_{n>=0} ( (1+x)^n - 1/(1+x)^n )^n. 3
1, 2, 15, 201, 3807, 93103, 2788528, 98816388, 4043274742, 187583369889, 9729671519992, 557914167187926, 35044465503390938, 2392988036211331477, 176493963957191423895, 13982630491776175877953, 1184241622895183679920962, 106774511855374079570593467, 10211007157153638802035266227, 1032332791948276849592811619207 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare to A319947, the dual to this sequence.

G.f. A(x) = (1+x) * B( x/(1+x) ), where B(x) is the g.f. of A319947.

a(n) - A319947(n) = 0 (mod 2) for n >= 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

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

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

a(n) ~ c * d^n * n! / sqrt(n), where d = 5.4666049332127684665699843922982444983683628264382802770893... and c = 0.3563391278539240852770166562386253680399190992740998... - Vaclav Kotesovec, Oct 10 2020

EXAMPLE

G.f.: A(x) = 1 + 2*x + 15*x^2 + 201*x^3 + 3807*x^4 + 93103*x^5 + 2788528*x^6 + 98816388*x^7 + 4043274742*x^8 + 187583369889*x^9 + ...

such that

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

Equivalently,

A(x) = 1  +

((1+x) - 1/(1+x))  +

((1+x)^4 - 2 + 1/(1+x)^4)  +

((1+x)^9 - 3*(1+x)^3 + 3/(1+x)^3 - 1/(1+x)^9)  +

((1+x)^16 - 4*(1+x)^8 + 6 - 4/(1+x)^8 + 1/(1+x)^16)  +

((1+x)^25 - 5*(1+x)^15 + 10*(1+x)^5 - 10/(1+x)^5 + 5/(1+x)^15 - 1/(1+x)^25)  +

((1+x)^36 - 6*(1+x)^24 + 15*(1+x)^12 - 20 + 15/(1+x)^12 - 6/(1+x)^24 + 1/(1+x)^36)  + ...

PROG

(PARI) {a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0, n, ((1+x)^m - 1/(1+X)^m)^m ); polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A319947.

Sequence in context: A042355 A208467 A221102 * A020557 A323118 A184361

Adjacent sequences:  A319463 A319464 A319465 * A319467 A319468 A319469

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 28 2018

STATUS

approved

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)