|
| |
|
|
A111184
|
|
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938.
|
|
0
| |
|
|
1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 34, 12, 1, 0, 120, 210, 110, 20, 1, 0, 720, 1452, 974, 270, 30, 1, 0, 5040, 11256, 8946, 3248, 560, 42, 1, 0, 40320, 97296, 87504, 38338, 8792, 1036, 56, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
FORMULA
| O.g.f. satisfies: A(x,y) = (1 + x^2*A'(x,y)) / (1+x - x*y - x*A(x,y)), where A'(x,y) = d/dx A(x,y). [From Paul D. Hanna, Jul 31 2011]
O.g.f. satisfies: A(x,y) = 1 - x * d/dx log(1+x - x*y - x*A(x,y)). [From Paul D. Hanna, Jul 30 2011]
Sum_{k, 0<=k<=n} T(n, k) = A003319(n+1).
Sum_{k, 0<=k<=n} T(n, k)*2^(n-k) = A004208(n).
|
|
|
EXAMPLE
| Rows begin:
1;
0, 1;
0, 2, 1;
0, 6, 6, 1;
0, 24, 34, 12, 1;
0, 120, 210, 110, 20, 1;
0, 720, 1452, 974, 270, 30, 1;
0, 5040, 11256, 8946, 3248, 560, 42, 1;
0, 40320, 97296, 87504, 38338, 8792, 1036, 56, 1.
|
|
|
PROG
| (PARI) {T(n, k)=local(A=1+x*y); for(i=1, n, A=1-x*deriv(log(1+x-x*y-x*A +x*O(x^n)))); polcoeff(polcoeff(A, n, x), k, y)} /* Paul D. Hanna */
(PARI) {T(n, k)=local(A=1+x*y); for(i=1, n, A=(1 + x^2*A')/(1 + x - x*y - x*A +x*O(x^n))); polcoeff(polcoeff(A, n, x), k, y)} /* Paul D. Hanna */
/* Print 10 Rows of the triangle: */
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
|
|
|
CROSSREFS
| Cf. A003319, A004208.
Sequence in context: A090238 A047922 A021830 * A111596 A129062 A163936
Adjacent sequences: A111181 A111182 A111183 * A111185 A111186 A111187
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Oct 16 2005
|
| |
|
|
