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%I #8 Jul 27 2019 21:41:50
%S 1,1,6,1,7,30,1,8,38,140,1,9,47,187,630,1,10,57,244,874,2772,1,11,68,
%T 312,1186,3958,12012,1,12,80,392,1578,5536,17548,51480,1,13,93,485,
%U 2063,7599,25147,76627,218790,1,14,107,592,2655,10254,35401,112028,330818
%N Infinite square array read by antidiagonals: a(q,n) is the coefficient of z^n in the series expansion of C(z)^q/(1-4z)^(3/2), where C(z) = (1-sqrt(1-4z))/(2z) is the Catalan function (q,n = 0,1,2,...).
%C a(q,n) = a(q-1,n) + a(q+1,n-1).
%C Row 0 is A002457; row 1 is A000531; row 2 is A029760; row 3 is A045720.
%F a(q,n) = Sum_{i=0..n} 4^i*binomial(2n-2i+q, n-i).
%e Array starts:
%e 1, 6, 30, 140, 630, ...
%e 1, 7, 38, 187, 874, ...
%e 1, 8, 47, 244, 1186, ...
%e 1, 9, 57, 312, 1578, ...
%e ...
%p a:=proc(q,n) options operator, arrow: sum(4^i*binomial(2*n-2*i+q, n-i), i= 0.. n) end proc: aa:=proc(q,n) options operator, arrow: a(q-1,n-1) end proc: matrix(10,10,aa); # yields sequence in matrix form
%Y Cf. A002457, A000531, A029760, A045720.
%K nonn,tabl
%O 0,3
%A _Emeric Deutsch_, Jul 24 2008
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