A128214 -id:A128214 - cp's OEIS frontend

A128213 Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2.

1, 1, 1, -1, -4, -4, 1, 7, 7, -1, -10, -10, 1, 13, 13, -1, -16, -16, 1, 19, 19, -1, -22, -22, 1, 25, 25, -1, -28, -28, 1, 31, 31, -1, -34, -34, 1, 37, 37, -1, -40, -40, 1, 43, 43, -1, -46, -46, 1, 49, 49, -1, -52, -52, 1, 55, 55, -1, -58, -58, 1, 61, 61, -1

Offset: 0

Paul Barry, Feb 19 2007

a(n+1) is the Hankel transform of {1,0,1,3,9,28,90,297,1001,3432,11934,...}, cf. A000245.
Binomial transform of A128214.
a(n+2) is the Hankel transform of A014138. - Paul Barry, Mar 15 2008

Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 0, 100}] (* Benedict W. J. Irwin, Nov 05 2016 *)

Vec((1-x+2*x^2-2*x^3)/(1-x+x^2)^2 + O(x^100)) \\ Michel Marcus, May 31 2014

a(n) = cos(Pi*n/3) + (2n/sqrt(3)-1/sqrt(3))*sin(Pi*n/3).

a(n) = y(n,n), where y(m+1,n) = y(m,n) - y(m-1,n), with y(0,n)=1 and y(1,n)=n. - Benedict W. J. Irwin, Nov 05 2016

A188126 Number of strictly increasing arrangements of 7 nonzero numbers in -(n+5)..(n+5) with sum zero.

Original entry on oeis.org

42, 152, 426, 1032, 2216, 4376, 8044, 13994, 23210, 37030, 57086, 85506, 124816, 178186, 249308, 342708, 463550, 618042, 813186, 1057238, 1359422, 1730468, 2182232, 2728362, 3383832, 4165678, 5092482, 6185216, 7466594, 8962070

Offset: 1

R. H. Hardin, Mar 21 2011

nonn

			Some solutions for n=6
-10..-10...-6...-7...-6..-11...-8..-10...-8..-11..-10...-9..-11..-11...-9...-9
.-9...-4...-3...-6...-5...-9...-7...-7...-7...-4...-7...-8...-9...-8...-6...-7
.-4...-2...-2...-4...-4...-3...-4...-6...-1...-3...-3...-3...-4...-4...-5...-4
..4....2...-1....1...-1...-1...-3...-1....1...-2...-1...-1....1....3...-4...-2
..5....3....1....3....3....4....5....6....3....1....1....5....2....4....7....4
..6....4....2....6....4....9....6....8....4....8....9....6...10....6....8....8
..8....7....9....7....9...11...11...10....8...11...11...10...11...10....9...10

R. H. Hardin, Table of n, a(n) for n = 1..200

Row 7 of A188122.

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+a(n-6)-2*a(n-7)+2*a(n-8)+a(n-9)-a(n-13)-2*a(n-14)+2*a(n-15)-a(n-16)+a(n-17)+a(n-19)-2*a(n-21)+a(n-22) =
   208637*n/12960 +413*(-1)^n/1152 +6403*n^3/1296 +355951*n^2/28800 +11*(-1)^n*n^2/384 +13*(-1)^n*n/96 +28669*n^4/25920 +709*n^5/5400 +841*n^6/129600 +6124649/777600 + (157*A049347(n)+74*A049347(n-1))/486 + 5*A128214(n+3)/81 +2*b(n)/25 + A057079(n+2)/18 -(-1)^(floor((n+1)/2))*A000034(n+1)/8 where b(n) is the 5-periodic sequence (-3,-1,-1,2,3,...) with offset 0.
Empirical: G.f. -2*x *(21 +34*x +61*x^2 +111*x^3 +152*x^4 +206*x^5 +217*x^6 +240*x^7 +212*x^8 +172*x^9 +120*x^10 +77*x^11 +36*x^12 +9*x^13 +11*x^14 -x^15 +4*x^16 +4*x^18 -8*x^20 +4*x^21) / ( (x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x+x^2)^2 *(1+x)^3 *(x-1)^7 ). - R. J. Mathar, Mar 21 2011

cp's OEIS Frontend

A128213 Expansion of (1-x+2x^2-2x^3)/(1-x+x^2)^2.

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