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A292741 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.

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