A110669 Sequence is {a(4,n)}, where a(m,n) is defined at sequence A110665.
0, 1, 4, 7, 4, -11, -38, -70, -100, -130, -172, -238, -328, -429, -528, -627, -744, -897, -1086, -1292, -1496, -1700, -1928, -2204, -2528, -2875, -3220, -3565, -3940, -4375, -4870, -5394, -5916, -6438, -6996, -7626, -8328, -9065, -9800, -10535, -11312, -12173, -13118, -14104, -15088, -16072, -17104
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 := 4: 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[4, n], {n, 0, 46}] (* Jean-François Alcover, Mar 29 2020 *)
Formula
Empirical g.f.: -x*(2*x-1) / ((x-1)^4*(x^2-x+1)^2). - Colin Barker, Jul 02 2014
Extensions
More terms from R. J. Mathar, Sep 01 2006