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 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. 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS 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) EXAMPLE {1}; {4,1}; {9,6,1}; {16,19,10,1}; {25,44,45,18,1}; ... 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 *) CROSSREFS Cf. A000290, A005900, A037270. Sequence in context: A197258 A211783 A185780 * A156308 A325004 A325012 Adjacent sequences:  A051669 A051670 A051671 * A051673 A051674 A051675 KEYWORD easy,nice,nonn,tabl AUTHOR Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de) STATUS approved

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Last modified February 18 12:10 EST 2020. Contains 332018 sequences. (Running on oeis4.)