A257673 - OEIS

%I #29 Oct 19 2022 11:14:42

%S 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 0,86,396,576,376,120,18,1,0,160,917,1719,1509,695,171,21,1,0,282,

%U 2036,4761,5340,3285,1158,231,24,1,0,500,4380,12441,17234,13473,6309,1792,300,27,1

%N 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 T is the convolution triangle of the number of plane partitions (A000219). - _Peter Luschny_, Oct 19 2022

%H Alois P. Heinz, <a href="/A257673/b257673.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255961(n,k-i).

%F G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j)^j)^k.

%e Triangle T(n,k) begins:

%e   1;

%e   0,   1;

%e   0,   3,    1;

%e   0,   6,    6,    1;

%e   0,  13,   21,    9,    1;

%e   0,  24,   62,   45,   12,    1;

%e   0,  48,  162,  174,   78,   15,    1;

%e   0,  86,  396,  576,  376,  120,   18,   1;

%e   0, 160,  917, 1719, 1509,  695,  171,  21,  1;

%e   0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1;

%e   ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, k*add(

%p       A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)

%p     end:

%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..12);

%p # Uses function PMatrix from A357368.

%p PMatrix(10, A000219); # _Peter Luschny_, Oct 19 2022

%t A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n];

%t T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)

%Y Columns k=0-10 give: A000007, A000219 (for n>0), A321947, A321948, A321949, A321950, A321951, A321952, A321953, A321954, A321955.

%Y Main diagonal and lower diagonals give: A000012, A008585, A081266.

%Y Row sums give A257674.

%Y T(2n,n) give A257675.

%Y Cf. A255961.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, May 03 2015