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A219121 Central terms in rows of triangle A219120. 2
1, 1, -2, -19, -74, -68, 1856, 22717, 182806, 1095506, 2706452, -62235754, -1630900556, -28213310474, -422792067164, -5747245586467, -68720160772442, -602550199498622, 1056275553274100, 251539588303778798, 9237652624016037908, 263685036472764512992 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The number of contiguous signs of the terms seems to grow roughly in proportion to the square-root of the number of terms.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..200

FORMULA

a(n) = [x^n] (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!.

EXAMPLE

Triangle A219120 begins:

1;

1, 1, -1;

1, 5, -2, -2, 1;

1, 15, 13, -19, 3, 3, -1;

1, 37, 128, -26, -74, 46, -4, -4, 1;

1, 83, 679, 755, -654, -68, 230, -90, 5, 5, -1;

1, 177, 2866, 9048, 2091, -5741, 1856, 498, -545, 155, -6, -6, 1; ...

in which the o.g.f. of row n, R(x,n), is given by:

R(x,n) = (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!;

note that the coefficient of x^n in R(x,n), for n>=1, forms this sequence.

The signs of the terms of this sequence begin:

+,+,

-,-,-,-,

+,+,+,+,+,

-,-,-,-,-,-,-,

+,+,+,+,+,+,+,+,+,+,

-,-,-,-,-,-,-,-,-,-,-,

+,+,+,+,+,+,+,+,+,+,+,+,+,

-,-,-,-,-,-,-,-,-,-,-,-,-,-,

+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,

-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,

+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,

-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,

+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+, ...

PROG

(PARI) {a(n)=polcoeff((1-x)^(2*n-1)*sum(k=0, 2*n, (k^n)*(k+1)^(k-1)*x^k/k!*exp(-(k+1)*x +x*O(x^n))), n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A219120.

Sequence in context: A024220 A024389 A110050 * A054209 A256112 A272053

Adjacent sequences:  A219118 A219119 A219120 * A219122 A219123 A219124

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 13 2012

STATUS

approved

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Last modified June 24 05:52 EDT 2021. Contains 345416 sequences. (Running on oeis4.)