A110670 Sequence is {a(5,n)}, where a(m,n) is defined at sequence A110665.
0, 1, 5, 12, 16, 5, -33, -103, -203, -333, -505, -743, -1071, -1500, -2028, -2655, -3399, -4296, -5382, -6674, -8170, -9870, -11798, -14002, -16530, -19405, -22625, -26190, -30130, -34505, -39375, -44769, -50685, -57123, -64119, -71745, -80073, -89138, -98938, -109473, -120785, -132958
Offset: 0
Examples
a(0,n): 0, 1, 0, -3, -4,... a(1,n): 0, 1, 1, -2, -6,... a(2,n): 0, 1, 2, 0, -6,... a(3,n): 0, 1, 3, 3, -3,... a(4,n): 0, 1, 4, 7, 4,... Main diagonal of array is 0, 1, 2, 3, 4,...
Programs
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Maple
A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end : nmax := 100 : m := 5: a := A11066x(m,nmax) : for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006 -
Mathematica
a[_, 0] = 0; a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))]; a[m_, n_] := a[m, n] = a[m-1, n] + a[m, n-1]; Table[a[5, n], {n, 0, 41}] (* Jean-François Alcover, Mar 29 2020 *)
Formula
Empirical g.f.: x*(2*x-1) / ((x-1)^5*(x^2-x+1)^2). - Colin Barker, Jul 02 2014
Extensions
More terms from R. J. Mathar, Sep 01 2006