A201509 Irregular triangle read by rows: T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k.

1, 1, 2, 2, 4, 5, 1, 8, 12, 4, 16, 28, 13, 1, 32, 64, 38, 6, 64, 144, 104, 25, 1, 128, 320, 272, 88, 8, 256, 704, 688, 280, 41, 1, 512, 1536, 1696, 832, 170, 10, 1024, 3328, 4096, 2352, 620, 61, 1, 2048, 7168

Offset

0,3

Comments

This is the pseudo-triangle whose successive lines are of the type T(n,0), T(n,1)+T(n-1,0), T(n,2)+T(n-1,1), ... T(n,k)+T(n-1,k-1), without 0's, with T=A201701. [e-mail, Philippe Deléham, Dec 04 2011]

Links

Table of n, a(n) for n=0..48.

Formula

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k. - Philippe Deléham, Dec 05 2011

The n-th row polynomial appears to equal Sum_{k = 1..floor((n+1)/2)} binomial(n,2*k-1)*(1+t)^k. Cf. A034867. - Peter Bala, Sep 10 2012

Aside from the first two rows below, the signed coefficients appear in the expansion (b*x - 1)^2 / (a*b*x^2 - 2a*x + 1) = 1 + (2 a - 2 b)x + (4 a^2 - 5 a b + b^2)x^2 + (8 a^3 - 12 a^2b + 4 ab^2)x^3 + ..., the reciprocal of the derivative of x*(1-a*x) / (1-b*x). This is related to A263633 via the expansion (a*b*x^2 - 2a*x + 1) / (b*x - 1)^2 = 1 + (b - a) (2x + 3b x^2 + 4b^2 x^3 + ...). See also A201780. - Tom Copeland, Oct 30 2023

Example

Triangle starts:
    1   1
    2   2
    4   5   1
    8  12   4
   16  28  13  1
   32  64  38  6
   64 144 104 25 1
  128 320 272 88 8
  ...
Triangle begins (full version):
    0
    1,   1
    2,   2,   0
    4,   5,   1,  0
    8,  12,   4,  0, 0
   16,  28,  13,  1, 0, 0
   32,  64,  38,  6, 0, 0, 0
   64, 144, 104, 25, 1, 0, 0, 0
  128, 320, 272, 88, 8, 0, 0, 0, 0

Crossrefs

Cf. A052542 (row sums).

Cf. A039991, A034867.

Cf. A201780, A263633.

Sequence in context: A049261 A135018 A073018 * A109295 A211188 A057899

Adjacent sequences: A201506 A201507 A201508 * A201510 A201511 A201512

Keyword

nonn,tabf

Author

Paul Curtz, Dec 02 2011

Extensions

Edited and new name using Philippe Deléham's formula, Joerg Arndt, Dec 13 2023

Status

approved