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%I #6 Mar 31 2012 13:20:06
%S 1,4,-6,4,-1,22,-80,139,-140,84,-28,4,140,-851,2500,-4536,5516,-4616,
%T 2640,-990,220,-22,969,-8420,35504,-94584,175564,-237600,239250,
%U -179960,100078,-40040,10920,-1820,140,7084,-80776,448056
%N Coefficient array for certain polynomials N(4; k,x) (rising powers in x).
%C The g.f. for the sequence of column r=3*k+j, k >= 0, j=1,2,3, of the staircase array A062750(n,r) is N(4; k,x)*(x^(k+1))/(1-x)^(3*k+1+j) with N(4; k,x) := sum(a(k,p)*x^p,p=0..3*k).
%C The m=0 column gives: A002293(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062752.
%C The sequence of step width of this staircase array is [1,3,3,3,...], i.e. the degree of the row polynomials is [0,3,6,9,...]= A008585.
%F a(k, p) := [x^p]N(4; k, x) with N(4; k, x)=(N(4; k-1, x)-A002293(k)*(1-x)^(3*k+1))/x, N(4; 0, x) := 1.
%F a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=0, .., (3*n-4); a(n, k)= ((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=(3*n-3), ..., 3*n; else 0.
%e {1}; {4,-6,4,-1}; {22,-80,139,-140,84,-28,4}; ...; N(4; 1,x)= 4-6*x+4*x^2-x^3 =(2-x)*(2-2*x+x^2).
%K sign,easy,tabf
%O 0,2
%A _Wolfdieter Lang_, Jul 12 2001
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