A219121 Central terms in rows of triangle A219120.

1, 1, -2, -19, -74, -68, 1856, 22717, 182806, 1095506, 2706452, -62235754, -1630900556, -28213310474, -422792067164, -5747245586467, -68720160772442, -602550199498622, 1056275553274100, 251539588303778798, 9237652624016037908, 263685036472764512992

Offset: 1

Author

Paul D. Hanna, Nov 13 2012

Keywords

• sign

Comments

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

Examples

			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:
+,+,
-,-,-,-,
+,+,+,+,+,
-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,
-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,
+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+, ...
		

Links

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

Crossrefs

Cf. A219120.

Programs

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),", "))

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!.