A177992 Triangle read by rows, A007318 * A177990.

1, 1, 1, 1, 3, 1, 1, 7, 3, 1, 1, 15, 6, 5, 1, 1, 31, 10, 16, 5, 1, 1, 63, 15, 42, 15, 7, 1, 1, 127, 21, 99, 35, 29, 7, 1, 1, 255, 28, 219, 70, 93, 28, 9, 1, 1, 511, 36, 466, 126, 256, 84, 46, 9, 1, 1, 1023, 45, 968, 210, 638, 210, 176, 45, 11, 1, 1, 2047, 55, 1981, 330, 1486, 462, 562, 165, 67, 11, 1

Offset: 0

Author

Gary W. Adamson, May 16 2010

Keywords

• nonn
• tabl

Comments

Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).

Double Riordan array ( 1/(1 - x); x/(1 - 2*x), x*(1 - 2*x)/(1 - x)^2 ) as defined in Davenport et al. - Peter Bala, Aug 25 2021

Examples

			First few rows of the triangle:
  1;
  1,    1;
  1,    3,  1;
  1,    7,  3,    1;
  1,   15,  6,    5,   1;
  1,   31, 10,   16,   5,    1;
  1,   63, 15,   42,  15,    7,   1;
  1,  127, 21,   99,  35,   29,   7,    1;
  1,  255, 28,  219,  70,   93,  28,    9,   1;
  1,  511, 36,  466, 126,  256,  84,   46,   9,   1;
  1, 1023, 45,  968, 210,  638, 210,  176,  45,  11,  1;
  1, 2047, 55, 1981, 330, 1486, 462,  562, 165,  67, 11,  1;
  1, 4095, 66, 4017, 495, 3302, 924, 1586, 495, 299, 66, 13, 1;
  ...
		

Links

• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Crossrefs

Cf. A177993 = A177990 * A007318.

Cf. A045623, A070909.

Formula

As infinite lower triangular matrices, A007318 * A177990.

From Peter Bala, Aug 25 2021: (Start)

T(n,2*k) = T(n-1,2*k-1) - T(n-1,2*k+1).

T(n,2*k+1) = 2*T(n-1,2*k+1) + T(n-1,2*k).

G.f.: A(x,t) = (1 - t)/(1 - 2*t)*(1 - 2*t + t*x)/((1 - t)^2 - t^2*x^2) = 1 + (1 + x)*t + (1 + 3*x + x^2)^t^2 + ....

G.f. column 2*k: x^(2*k)/(1 - x)^(2*k+1).

G.f. column 2*k+1: x^(2*k+1)/((1 - x)^(2*k+1) * (1 - 2*x)). (End)

Extensions

a(8) corrected and more terms by Georg Fischer, Dec 28 2021