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A330787 Triangle read by rows: T(n,k) is the number of strict multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

%I A330787 #12 Dec 17 2020 14:47:10
%S A330787 1,2,1,4,8,1,8,32,18,1,16,124,140,32,1,32,444,888,432,50,1,64,1568,
%T A330787 5016,4196,1060,72,1,128,5440,26796,34732,15064,2224,98,1,256,18768,
%U A330787 138292,262200,174240,44348,4172,128,1,512,64432,698864,1870840,1781884,692668,112424,7200,162,1
%N A330787 Triangle read by rows: T(n,k) is the number of strict multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.
%H A330787 Andrew Howroyd, <a href="/A330787/b330787.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A330787 Triangle begins:
%e A330787     1;
%e A330787     2,    1;
%e A330787     4,    8,     1;
%e A330787     8,   32,    18,     1;
%e A330787    16,  124,   140,    32,     1;
%e A330787    32,  444,   888,   432,    50,    1;
%e A330787    64, 1568,  5016,  4196,  1060,   72,  1;
%e A330787   128, 5440, 26796, 34732, 15064, 2224, 98, 1;
%e A330787   ...
%e A330787 The T(3,1) = 4 multiset partitions are {{1,1,1}}, {{1,1,2}}, {{1,2,2}}, {{1,2,3}}.
%e A330787 The T(3,2) = 8 multiset partitions are {{1},{1,1}}, {{1},{2,2}}, {{2},{1,2}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
%e A330787 The T(3,3) = 1 multiset partition is {{1},{2},{3}}.
%t A330787 B[n_, k_] := Sum[Binomial[r, k] (-1)^(r-k), {r, k, n}];
%t A330787 row[n_] := Sum[B[n, j] SeriesCoefficient[ Product[(1 + x^k y)^Binomial[k + j - 1, j - 1], {k, 1, n}], {x, 0, n}], {j, 1, n}]/y + O[y]^n // CoefficientList[#, y]&;
%t A330787 Array[row, 10] // Flatten (* _Jean-François Alcover_, Dec 17 2020, after _Andrew Howroyd_ *)
%o A330787 (PARI) \\ here B(n, k) is A239473(n, k)
%o A330787 B(n,k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
%o A330787 Row(n)={Vecrev(sum(j=1, n, B(n,j)*polcoef(prod(k=1, n, (1 + x^k*y + O(x*x^n))^binomial(k+j-1,j-1)), n))/y)}
%o A330787 { for(n=1, 10, print(Row(n))) }
%Y A330787 Row sums are A317776.
%Y A330787 Column 1 is A000079(n-1).
%Y A330787 Main diagonal is A000012.
%Y A330787 Cf. A317532, A327116.
%K A330787 nonn,tabl
%O A330787 1,2
%A A330787 _Andrew Howroyd_, Dec 31 2019