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%I #8 Nov 21 2013 13:11:32
%S 1,4,1,9,6,1,16,19,10,1,25,44,45,18,1,36,85,136,115,34,1,49,146,325,
%T 452,309,66,1,64,231,666,1333,1576,859,130,1,81,344,1225,3254,5725,
%U 5684,2445,258,1,100,489,2080,6951,16626,25405,21016,7075,514,1,121
%N Triangle of up-down sums of k-th powers: a(n,k)=sum(i^k,i=1..n)+sum((n-i)^k,i=1..n-1), n,k>0.
%F a(n, 1)=n^2=A000290, a(n, 2)=1/3*n*(2*n^2+1)=A005900, a(n, 3)= (1/2) *n^2*(n^2+1)=A037270, a(n, 4)=1/15*n*(6*n^4+10*n^2-1), a(n, 5)=1/6*n^2*(2*n^4+5*n^2-1)
%e {1}; {4,1}; {9,6,1}; {16,19,10,1}; {25,44,45,18,1}; ...
%t a[n_, k_] := HarmonicNumber[n, -k]+Zeta[-k]-Zeta[-k, n]; Flatten[ Table[ a[n-k+1, k], {n, 1, 11}, {k, 1, n}]] (* _Jean-François Alcover_, Nov 29 2011 *)
%Y Cf. A000290, A005900, A037270.
%K easy,nice,nonn,tabl
%O 1,2
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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