A086990 -id:A086990 - cp's OEIS frontend

A090412 A Chebyshev transform of 2^n.

1, 2, 3, 4, 6, 10, 15, 20, 30, 52, 78, 96, 144, 282, 423, 420, 630, 1660, 2490, 1304, 1956, 11332, 16998, -3896, -5844, 95240, 142860, -157160, -235740, 983610, 1475415, -2634300, -3951450, 11751660, 17627490, -38381160, -57571740, 152461740, 228692610

Offset: 0

Paul Barry, Dec 05 2003

Cf. A000108, A086990, A114710.

c(x) = (1 - sqrt(1 - 4*x)) / (2*x);
my(x='x+O('x^40)); Vec(c(-x^2)/(1-2*x*c(-x^2))) \\ Michel Marcus, Feb 06 2022

G.f.: A(x) = c(-x^2)/(1-2*x*c(-x^2)), c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{k=0..n} (k+1)*binomial(n, n/2-k/2)*(-1)^(n/2 - k/2)*(1 + (-1)^(n+k))*2^k/(n+k+2).
Let M be a tridiagonal matrix with 1's in the superdiagonal, [1,0,0,0,...] in the main diagonal, and [1,-1,-1,-1,...] in the subdiagonal; and V = vector [1,0,0,0,...]. The sequence is generated as a left column using iterates of M^n*V. - Gary W. Adamson, Jun 08 2011
D-finite with recurrence 2*(n+1)*a(n) -3*(n+1)*a(n-1) +8(n-2)*a(n-2) +12*(2-n)*a(n-3)=0. - R. J. Mathar, Nov 09 2012
The o.g.f. A(x) = (1/x) * Series reversion of x*(1 + 2*x)/((1 + x)*(1 + 3*x)). - Peter Bala, Nov 07 2022

A375025 Triangle read by rows: Matrix inverse of row-reversed A374439.

Original entry on oeis.org

1, -2, 1, 3, -2, 1, -4, 2, -2, 1, 6, -2, 1, -2, 1, -10, 5, 0, 0, -2, 1, 15, -10, 5, 2, -1, -2, 1, -20, 10, -12, 6, 4, -2, -2, 1, 30, -8, 4, -16, 8, 6, -3, -2, 1, -52, 26, 8, -4, -22, 11, 8, -4, -2, 1, 78, -60, 30, 30, -15, -30, 15, 10, -5, -2, 1

Offset: 0

Peter Luschny, Aug 07 2024

			Triangle starts:
  [0] [  1]
  [1] [ -2,   1]
  [2] [  3,  -2,   1]
  [3] [ -4,   2,  -2,   1]
  [4] [  6,  -2,   1,  -2,   1]
  [5] [-10,   5,   0,   0,  -2,  1]
  [6] [ 15, -10,   5,   2,  -1, -2,  1]
  [7] [-20,  10, -12,   6,   4, -2, -2,  1]
  [8] [ 30,  -8,   4, -16,   8,  6, -3, -2,  1]
  [9] [-52,  26,   8,  -4, -22, 11,  8, -4, -2, 1]

Column 0 and row sums: A086990, A090412; alternating row sums: A375026.

Cf. A374439.

A := (n,k) -> ifelse(k::odd,2,1)*binomial(n-irem(k,2)-iquo(k,2),iquo(k,2)):
ARevRow := n -> local k; [seq(A(n, n-k), k = 0..n)]:
M := m -> Matrix(m, (n, k) -> ifelse(k > n, 0, ARevRow(n-1)[k])):
T := n -> LinearAlgebra:-MatrixInverse(M(n)): T(11);

from functools import cache
@cache
def Trow(n):
    if n == 0: return [1]
    if n == 1: return [-2, 1]
    fli = Trow(n - 1)
    row = [1] * (n + 1)
    row[n - 1] = -2
    for k in range(n - 2, 0, -1):
        row[k] = fli[k - 1] - fli[k + 1]
    row[0] = -2 * fli[0] - fli[1]
    return row
# Peter Luschny, Aug 18 2024

A094184 Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 1, -1, 1, 4, 3, -2, 0, 1, 5, 6, -2, -2, 2, 1, 6, 10, 0, -6, 4, 0, 1, 7, 15, 5, -11, 3, 5, -5, 1, 8, 21, 14, -15, -4, 15, -10, 0, 1, 9, 28, 28, -15, -19, 26, -6, -14, 14, 1, 10, 36, 48, -7, -42, 30, 16, -42, 28, 0, 1, 11, 45, 75, 14, -70, 16, 60, -70, 14, 42, -42, 1, 12, 55, 110, 54, -96, -28, 120, -70, -56, 126, -84, 0, 1

Offset: 0

Wouter Meeussen, May 06 2004

Row sums are A086990 or A090412. (Superseeker finds that the j-th coefficient of OGF(A090412)(z)*(1-z)^j equals A049122). Same absolute values as A065432. Even rows end in 0, odd rows end in Catalan numbers (A000118) with alternating sign.

			Table starts {1},{1,1},{1,2,0},{1,3,1,-1},{1,4,3,-2,0},{1,5,6,-2,-2,2}

Cf. A086990, A090412, A049122, A009766, A065432, A065441.

T[, 0]:=1;T[0, 0]:=1;T[i, j_]/;j>i:=0;T[i_, j_]:=T[i, j]=T[i-1, j]-T[i, j-1]+2 T[i-1, j-1]

T(i, j)=T(i-1, j)-T(i, j-1)+2*T(i-1, j-1), with T(i, 0)=1 and T(i, j)=0 if j>i.

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A090412 A Chebyshev transform of 2^n.

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A375025 Triangle read by rows: Matrix inverse of row-reversed A374439.

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A094184 Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.

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