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%I #6 Mar 31 2012 13:20:06
%S 1,5,-10,10,-5,1,35,-170,415,-629,630,-420,180,-45,5,285,-2315,9381,
%T -24395,44625,-59880,60015,-45040,25025,-10010,2730,-455,35,2530,
%U -29379,169405,-633675,1703700,-3467145,5497640,-6903325
%N Coefficient array for certain polynomials N(5; k,x) (rising powers in x).
%C The g.f. for the sequence of column r=4*k+j, k >= 0, j=1,2,3,4, of the staircase array A062985(n,r) is N(5; k,x)*(x^(k+1))/(1-x)^(4*k+1+j) with N(5; k,x) := sum(a(k,p)*x^p,p=0..4*k).
%C The m=0 column gives A002294(k+1). The row sums give A000012 (powers of 1) and (unsigned) A062987.
%C The sequence of step width of this staircase array is [1,4,4,4,...], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.
%F a(k, p) := [x^p]N(5; k, x) with N(5; k, x)=(N(5; k-1, x)- A002294(k)*(1-x)^(4*k+1))/x, N(5; 0, x) := 1.
%F a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=0, .., (4*n-5); a(n, k)= ((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=(4*n-4), ..., 4*n; else 0.
%e {1}; {5,-10,10,-5,1}; {35,-170,415,-629,630,-420,180,-45,5}; ...; N(5; 1,x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x.
%Y A062991, A062746, A062751.
%K sign,easy,tabf
%O 0,2
%A _Wolfdieter Lang_, Jul 12 2001
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