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A177992 Triangle read by rows, A007318 * A177990. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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
LINKS
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
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)
EXAMPLE
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;
...
CROSSREFS
Sequence in context: A332538 A071812 A248133 * A346906 A228524 A116407
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 16 2010
EXTENSIONS
a(8) corrected and more terms by Georg Fischer, Dec 28 2021
STATUS
approved

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Last modified March 1 08:00 EST 2024. Contains 370430 sequences. (Running on oeis4.)