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%I #15 May 08 2022 22:49:15
%S 0,1,3,9,1,28,7,90,35,1,297,154,11,1001,637,77,1,3432,2548,440,15,
%T 11934,9996,2244,135,1,41990,38760,10659,950,19,149226,149226,48279,
%U 5775,209,1,534888,572033,211508,31878,1748,23,1931540,2187185,904475,164450
%N Triangle read by rows: T(n,k) = (4k+3)/(n+2k+2)*binomial(2n,n+2k+1).
%C This triangle forms a companion to A119245.
%C Combinatorial interpretations of T(n,k):
%C 1) The number of standard tableaux of shape (n-2*k-1,n+2*k+1).
%C 2) The entries in column k are (with an offset of 2*k+1) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k+2. See [Sunik, Theorem 4].
%H Zoran Sunic, <a href="https://doi.org/10.37236/1745">Self describing sequences and the Catalan family tree</a>, Elect. J. Combin., 10 (No. 1, 2003).
%F T(n,k) = (4*k+3)/(n+2*k+2)*binomial(2*n,n+2*k+1).
%F O.g.f. y*C(y)^3/(1 - x*y^2*C(y)^4) = y + 3*y^2 + (9 + x)*y^3 + (28 + 7*x)*y^4 + ..., where C(x) = [1-(1-4*x)^(1/2)]/(2*x) is the o.g.f. for the Catalan numbers A000108.
%F Row sums A001700.
%e Triangle begins
%e ==================================
%e n\k|.....0.....1.....2.....3.....4
%e ==================================
%e .0.|.....0
%e .1.|.....1
%e .2.|.....3
%e .3.|.....9.....1
%e .4.|....28.....7
%e .5.|....90....35.....1
%e .6.|...297...154....11
%e .7.|..1001...637....77.....1
%e .8.|..3432..2548...440....15
%e .9.|.11934..9996..2244...135.....1
%p with(combinat): T:=(n,k) -> (4k+3)/(n+2k+2)*binomial(2n,n+2k+1): for n from 0 to 13 do seq(T(n,k),k = 0..6); end do;
%Y Cf. A000245 (column 0), A000588 (column 1), A000589 (column 2), A001700 (row sums), A119245.
%K easy,nonn,tabf
%O 0,3
%A _Peter Bala_, Mar 20 2009
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