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A257673 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.

%I A257673 #29 Oct 19 2022 11:14:42
%S A257673 1,0,1,0,3,1,0,6,6,1,0,13,21,9,1,0,24,62,45,12,1,0,48,162,174,78,15,1,
%T A257673 0,86,396,576,376,120,18,1,0,160,917,1719,1509,695,171,21,1,0,282,
%U A257673 2036,4761,5340,3285,1158,231,24,1,0,500,4380,12441,17234,13473,6309,1792,300,27,1
%N A257673 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.
%C A257673 T is the convolution triangle of the number of plane partitions (A000219). - _Peter Luschny_, Oct 19 2022
%H A257673 Alois P. Heinz, <a href="/A257673/b257673.txt">Rows n = 0..140, flattened</a>
%F A257673 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255961(n,k-i).
%F A257673 G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j)^j)^k.
%e A257673 Triangle T(n,k) begins:
%e A257673   1;
%e A257673   0,   1;
%e A257673   0,   3,    1;
%e A257673   0,   6,    6,    1;
%e A257673   0,  13,   21,    9,    1;
%e A257673   0,  24,   62,   45,   12,    1;
%e A257673   0,  48,  162,  174,   78,   15,    1;
%e A257673   0,  86,  396,  576,  376,  120,   18,   1;
%e A257673   0, 160,  917, 1719, 1509,  695,  171,  21,  1;
%e A257673   0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1;
%e A257673   ...
%p A257673 A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
%p A257673       A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
%p A257673     end:
%p A257673 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A257673 seq(seq(T(n, k), k=0..n), n=0..12);
%p A257673 # Uses function PMatrix from A357368.
%p A257673 PMatrix(10, A000219); # _Peter Luschny_, Oct 19 2022
%t A257673 A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
%t A257673 T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A257673 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)
%Y A257673 Columns k=0-10 give: A000007, A000219 (for n>0), A321947, A321948, A321949, A321950, A321951, A321952, A321953, A321954, A321955.
%Y A257673 Main diagonal and lower diagonals give: A000012, A008585, A081266.
%Y A257673 Row sums give A257674.
%Y A257673 T(2n,n) give A257675.
%Y A257673 Cf. A255961.
%K A257673 nonn,tabl
%O A257673 0,5
%A A257673 _Alois P. Heinz_, May 03 2015