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A067327 Triangle related to generalized Catalan numbers A064340. 1
1, 1, 3, 4, 12, 12, 28, 84, 96, 48, 256, 768, 912, 576, 192, 2704, 8112, 9792, 6720, 3072, 768, 31168, 93504, 113856, 81408, 42240, 15360, 3072, 380608, 1141824, 1397760, 1023744, 568320, 242688, 73728 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The row polynomials Z(2,2; n,y)= sum(a(n,m)*y^m,m=0..n) appear in c(2,2; x) (the g.f. of C(2,2; n) := A064340(n)) with the first (n+1) expansion terms subtracted, as follows: c(2,2; x)-sum(C(2,2; k)*x^k,k=0..n) = x^(n+1)*G(2,2; x)*Z(2,2; n,y), n>=0, where y=c(4*x) and c(x) is the g.f. of A000108 (Catalan) and G(2,2; x) is the g.f. of C(2,2; n+1), that is G(2,2; x)= (c(2,2; x)-1)/x. Hence G(2,2; x)*Z(2,2; k,c(4*x)) is, for k=0,1,..., the g.f. for C(2,2; n+k), n>=0.
Column sequences are: A064340(n), 3*A064340(n+1), Main diagonal gives A002001(n). Row sums give C(2,2; n+1)= A064340(n+1).
LINKS
FORMULA
a(n, 0)= C(2, 2; n) := A064340(n), n>=0; a(n, 1)= 3*C(2, 2; n), n>=1; a(n, m)=4*sum(a(n-1, k), k=(m-1)..(n-1)) if n>=m>=2, else 0.
CROSSREFS
Cf. A067328 (scaled triangle with 1's in main diagonal).
Sequence in context: A348996 A196440 A101404 * A346470 A218968 A332206
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Feb 05 2002
STATUS
approved

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)