|
| |
|
|
A111049
|
|
Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,0, ...] where DELTA is the operator defined in A084938.
|
|
1
|
|
|
|
1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 11, 27, 25, 8, 1, 20, 70, 100, 65, 16, 1, 37, 170, 330, 325, 161, 32, 1, 70, 399, 980, 1295, 966, 385, 64, 1, 135, 917, 2723, 4515, 4501, 2635, 2695, 897, 128, 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
Table of n, a(n) for n=0..54.
|
|
|
FORMULA
|
T(n, k) = 2^(n-1)binomial(n-1, k-1)+binomial(n-1, k).
Sum_{k, 0<=k<=n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n>=1.
From Peter Bala, Mar 20 2013: (Start)
O.g.f. : (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....
E.g.f. : (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....
Recurrence equation: for n >= 1, T(n+1,k) = 2*T*n,k) + 2*T(n,k-1) - binomial(n,k). (End)
|
|
|
EXAMPLE
|
Rows begin:
1;
1, 1;
1, 3, 2;
1, 6, 9, 4;
1, 11, 27, 25, 8;
1, 20, 70, 100, 65, 16;
1, 37, 170, 330, 325, 161, 32;
1, 70, 399, 980, 1295, 966, 385, 64;
1, 135, 917, 2723, 4515, 4501, 2695, 897, 128;
1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;
|
|
|
CROSSREFS
|
Cf. A002064, A006127.
Sequence in context: A052174 A181897 A212207 * A211955 A088617 A190909
Adjacent sequences: A111046 A111047 A111048 * A111050 A111051 A111052
|
|
|
KEYWORD
|
easy,nonn,tabl
|
|
|
AUTHOR
|
Philippe DELEHAM, Oct 07 2005
|
|
|
STATUS
|
approved
|
| |
|
|
