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%I #29 Oct 26 2018 17:05:08
%S 1,1,0,1,1,1,1,2,2,1,1,3,5,3,2,1,4,10,11,5,2,1,5,17,31,24,7,4,1,6,26,
%T 69,95,50,11,4,1,7,37,131,278,287,104,15,7,1,8,50,223,657,1114,865,
%U 212,22,8,1,9,65,351,1340,3287,4460,2599,431,30,12,1,10,82,521,2459,8042,16439,17844,7804,870,42,14
%N Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A292741/b292741.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).
%F A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).
%e A(1,3) = 3: 1a, 1b, 1c.
%e A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.
%e A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 1, 2, 5, 10, 17, 26, 37, 50, ...
%e 1, 3, 11, 31, 69, 131, 223, 351, ...
%e 2, 5, 24, 95, 278, 657, 1340, 2459, ...
%e 2, 7, 50, 287, 1114, 3287, 8042, 17215, ...
%e 4, 11, 104, 865, 4460, 16439, 48256, 120509, ...
%e 4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...
%p b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,
%p add(b(n-i*j, i-1, k), j=0..iquo(n, i)))
%p end:
%p A:= (n, k)-> b(n$2, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[0, _, _] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]];
%t A[n_, k_] := b[n, n, k];
%t Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 19 2018, translated from Maple *)
%Y Columns k=0-2 give: A002865, A000041, A090764.
%Y Rows n=0-2 give: A000012, A001477, A002522, A071568.
%Y Main diagonal gives A292462.
%Y Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 22 2017
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