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A003988 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].

%I A003988 #33 Jul 02 2025 16:01:54
%S A003988 1,2,0,3,1,0,4,1,0,0,5,2,1,0,0,6,2,1,0,0,0,7,3,1,1,0,0,0,8,3,2,1,0,0,
%T A003988 0,0,9,4,2,1,1,0,0,0,0,10,4,2,1,1,0,0,0,0,0,11,5,3,2,1,1,0,0,0,0,0,12,
%U A003988 5,3,2,1,1,0,0,0,0,0,0,13,6,3,2,1,1,1,0,0,0,0,0,0,14,6,4,2,2,1,1,0,0,0,0
%N A003988 Triangle with subscripts (1,1),(2,1),(1,2),(3,1),(2,2),(1,3), etc. in which entry (i,j) is [ i/j ].
%C A003988 Another version of A010766.
%H A003988 Reinhard Zumkeller, <a href="/A003988/b003988.txt">Table of n, a(n) for n = 1..5050</a>
%F A003988 From _Franklin T. Adams-Watters_, Jan 28 2006: (Start)
%F A003988 T(n,k) = Sum_{i=1..k} A077049(n,i).
%F A003988 G.f.: (1/(1-x))*Sum_{k>0} x^k*y^k/(1-x^k) = (1/(1-x))*Sum_{k>0} x^k * y / (1 - x^k y) = (1/(1-x)) * Sum_{k>0} x^k * Sum_{d|k} y^d = A(x,y)/(1-x) where A(x,y) is the g.f. of A077049. (End)
%F A003988 T(n,k) = floor((n + 1 - k) / k). - _Reinhard Zumkeller_, Apr 13 2012
%t A003988 t[n_, k_] := Quotient[n, k]; Table[t[n-k+1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 21 2013 *)
%o A003988 (Haskell)
%o A003988 a003988 n k = (n + 1 - k) `div` k
%o A003988 a003988_row n = zipWith div [n,n-1..1] [1..n]
%o A003988 a003988_tabl = map a003988_row [1..]
%o A003988 -- _Reinhard Zumkeller_, Apr 13 2012
%Y A003988 Cf. A010766, A003056, A049581, A003991, A004247, A077049.
%Y A003988 Row sums are in A006218. Antidiagonal sums are in A002541.
%K A003988 tabl,nonn,easy,nice
%O A003988 1,2
%A A003988 _Marc LeBrun_
%E A003988 More terms from _James Sellers_