A051672 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.

1, 4, 1, 9, 6, 1, 16, 19, 10, 1, 25, 44, 45, 18, 1, 36, 85, 136, 115, 34, 1, 49, 146, 325, 452, 309, 66, 1, 64, 231, 666, 1333, 1576, 859, 130, 1, 81, 344, 1225, 3254, 5725, 5684, 2445, 258, 1, 100, 489, 2080, 6951, 16626, 25405, 21016, 7075, 514, 1, 121

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Keywords

• easy
• nice
• nonn
• tabl

Examples

			{1}; {4,1}; {9,6,1}; {16,19,10,1}; {25,44,45,18,1}; ...
		

Crossrefs

Cf. A000290, A005900, A037270.

Programs

Mathematica

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 *)

Formula

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)