

A118343


Triangle, read by rows, where diagonals are successive selfconvolutions of A108447.


3



1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 20, 0, 1, 4, 15, 48, 113, 0, 1, 5, 22, 85, 282, 688, 0, 1, 6, 30, 132, 519, 1762, 4404, 0, 1, 7, 39, 190, 837, 3330, 11488, 29219, 0, 1, 8, 49, 260, 1250, 5516, 22135, 77270, 199140, 0, 1, 9, 60, 343, 1773, 8461, 37404, 151089
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OFFSET

0,8


COMMENTS

A108447 equals the central terms of pendular triangle A118340 and the diagonals of this triangle form the semidiagonals of the triangle A118340. Row sums equal A054727, the number of forests of rooted trees with n nodes on a circle without crossing edges.


LINKS

Table of n, a(n) for n=0..62.


FORMULA

Since g.f. G=G(x) of A108447 satisfies: G = 1  x*G + x*G^2 + x*G^3 then T(n,k) = T(n1,k)  T(n1,k1) + T(n,k1) + T(n+1,k1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n1,k) + Sum_{j=1..k} [2*T(n1+j,kj)  T(n2+j,kj)] for n>k>=0.


EXAMPLE

Show: T(n,k) = T(n1,k)  T(n1,k1) + T(n,k1) + T(n+1,k1)
at n=8,k=4: T(8,4) = T(7,4)  T(7,3) + T(8,3) + T(9,3)
or 837 = 519  132 + 190 + 260.
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 2, 4, 0;
1, 3, 9, 20, 0;
1, 4, 15, 48, 113, 0;
1, 5, 22, 85, 282, 688, 0;
1, 6, 30, 132, 519, 1762, 4404, 0;
1, 7, 39, 190, 837, 3330, 11488, 29219, 0;
1, 8, 49, 260, 1250, 5516, 22135, 77270, 199140, 0;
1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 0; ...


PROG

(PARI) {T(n, k)=polcoeff((serreverse(x*(1x+sqrt((1x)*(15*x)+x*O(x^k)))/2/(1x))/x)^(nk), k)}


CROSSREFS

Cf. A108447, A054727 (row sums), A118340.
Sequence in context: A258761 A256245 A173004 * A309148 A226031 A308460
Adjacent sequences: A118340 A118341 A118342 * A118344 A118345 A118346


KEYWORD

nonn,tabl


AUTHOR

Paul D. Hanna, Apr 26 2006


STATUS

approved



