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%I #44 Dec 05 2023 16:07:09
%S 1,1,1,2,3,2,3,7,7,3,5,14,21,14,5,7,26,48,48,26,7,11,45,103,131,103,
%T 45,11,15,75,198,312,312,198,75,15,22,120,366,674,830,674,366,120,22,
%U 30,187,637,1359,1961,1961,1359,637,187,30,42,284,1078,2584,4302,5066,4302,2584,1078,284,42
%N Euler transform of Pascal's triangle A007318.
%C Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - _Franklin T. Adams-Watters_, Jan 10 2007
%H Alois P. Heinz, <a href="/A055375/b055375.txt">Rows n = 0..200, flattened</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - _Franklin T. Adams-Watters_, Jan 10 2007
%F Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - _Alois P. Heinz_, Dec 05 2023
%e Triangle begins
%e 1;
%e 1, 1;
%e 2, 3, 2;
%e 3, 7, 7, 3;
%e 5, 14, 21, 14, 5;
%e 7, 26, 48, 48, 26, 7;
%e 11, 45, 103, 131, 103, 45, 11;
%e 15, 75, 198, 312, 312, 198, 75, 15;
%e ...
%p g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
%p end:
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p end:
%p T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..15); # _Alois P. Heinz_, Feb 14 2023
%t nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 18 2017 *)
%Y Row sums give A034899.
%Y Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
%Y T(2n,n) gives A360626.
%Y Cf. A007318, A034691, A209406, A360634.
%K nonn,tabl
%O 0,4
%A _Christian G. Bower_, May 16 2000
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