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 A204459 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of k-element subsets that can be chosen from {1,2,...,k*n} having element sum k*(k*n+1)/2. 23
 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 3, 0, 1, 0, 1, 8, 8, 4, 1, 1, 0, 1, 0, 33, 0, 5, 0, 1, 0, 1, 58, 141, 86, 25, 6, 1, 1, 0, 1, 0, 676, 0, 177, 0, 7, 0, 1, 0, 1, 526, 3370, 3486, 1394, 318, 50, 8, 1, 1, 0, 1, 0, 17575, 0, 11963, 0, 519, 0, 9, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A(n,k) is the number of partitions of k*(k*n+1)/2 into k distinct parts <=k*n. A(n,k) = 0 if k>0 and (n = 0 or k*(k*n+1) mod 2 = 1). LINKS Alois P. Heinz, Antidiagonals d=0..60 EXAMPLE A(0,0) = 1: {}. A(1,1) = 1: {1}. A(5,1) = 1: {3}. A(1,5) = 1: {1,2,3,4,5}. A(2,2) = 2: {1,4}, {2,3}. A(3,2) = 3: {1,6}, {2,5}, {3,4}. A(2,3) = 0: no subset of {1,2,3,4,5,6} has element sum 3*(3*2+1)/2 = 21/2. A(4,2) = 4: {1,8}, {2,7}, {3,6}, {4,5}. A(3,3) = 8: {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, {4,5,6}. A(2,4) = 8: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}. Square array A(n,k) begins: 1, 0, 0, 0, 0, 0, 0, 0, ... 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 0, 2, 0, 8, 0, 58, 0, ... 1, 1, 3, 8, 33, 141, 676, 3370, ... 1, 0, 4, 0, 86, 0, 3486, 0, ... 1, 1, 5, 25, 177, 1394, 11963, 108108, ... 1, 0, 6, 0, 318, 0, 32134, 0, ... 1, 1, 7, 50, 519, 5910, 73294, 957332, ... MAPLE b:= proc(n, i, t) option remember; `if`(it*(2*i-t+1)/2, 0, `if`(n=0, 1, b(n, i-1, t) +`if`(nt*((2*i-t+1)/2) = 0; b[0, _, _] = 1; b[n_, i_, t_] := b[n, i, t] = b[n, i-1, t] + If[n

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Last modified September 21 02:47 EDT 2023. Contains 365486 sequences. (Running on oeis4.)