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A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

%I A069270 #27 Jan 05 2025 19:51:36
%S A069270 1,1,1,1,2,4,1,3,9,22,1,4,15,52,140,1,5,22,91,340,969,1,6,30,140,612,
%T A069270 2394,7084,1,7,39,200,969,4389,17710,53820,1,8,49,272,1425,7084,32890,
%U A069270 135720,420732,1,9,60,357,1995,10626,53820,254475,1068012,3362260
%N A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).
%C A069270 For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).
%C A069270 Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - _Tom Copeland_, Dec 25 2019
%H A069270 Michael De Vlieger, <a href="/A069270/b069270.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)
%H A069270 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%F A069270 T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
%F A069270 For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
%F A069270 T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).
%e A069270 Rows start
%e A069270   1;
%e A069270   1,   1;
%e A069270   1,   2,   4;
%e A069270   1,   3,   9,  22;
%e A069270   1,   4,  15,  52, 140;
%e A069270 etc.
%p A069270 A069270 := proc(n,k)
%p A069270         binomial(n+3*k,k)*(n-k+1)/(n+2*k+1) ;
%p A069270 end proc: # _R. J. Mathar_, Oct 11 2015
%t A069270 Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 27 2019 *)
%Y A069270 Columns include A000012, A000027, A055999.
%Y A069270 Right-hand diagonals include A002293, A069271, A006632.
%Y A069270 Cf. A130458 (row sums).
%K A069270 nonn,tabl,easy
%O A069270 0,5
%A A069270 _Henry Bottomley_, Mar 12 2002