cp's OEIS Frontend

A109876 Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = Sum_{i=0..n-1} Product_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.

%I A109876 #15 Jan 09 2021 10:07:52
%S A109876 1,3,2,6,11,6,10,24,50,24,15,45,120,274,120,21,76,252,720,1764,720,28,
%T A109876 119,476,1680,5040,13068,5040,36,176,828,3520,12960,40320,109584,
%U A109876 40320,45,249,1350,6750,29880,113400,362880,1026576,362880,55,340,2090,12048
%N A109876 Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = Sum_{i=0..n-1} Product_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.
%C A109876 The first four columns (excluding the initial term of each) are A000217 (triangular numbers), A006527, A062026 and A062027. The first and third diagonals are both A000142 (factorials). The second diagonal is A000254.
%C A109876 Without the exception for k = n, a(n, n) would be n*n! (A001563(n)). For example, a(3, 3) would be 1*2*3 + 2*3*1 + 3*1*2 instead of 1*2*3. The author's original description did not mention the exception. I guess it didn't make sense to him to add n identical terms. - _David Wasserman_, Oct 01 2008
%e A109876 a(5, 3) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 + 5*1*2 = 120.
%o A109876 (PARI) f(x, y) = if (x > y, x - y, x);
%o A109876 a(n, k) = if (n == k, n!, sum (i = 0, n - 1, prod (j = i + 1, i + k, f(j, n)))); \\ _David Wasserman_, Oct 01 2008
%Y A109876 Cf. A109877.
%K A109876 nonn,easy,tabl
%O A109876 1,2
%A A109876 _Amarnath Murthy_, Jul 10 2005
%E A109876 Edited and extended by _David Wasserman_, Oct 01 2008