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 A360764 Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows. 4
 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 1, 4, 6, 1, 0, 1, 6, 8, 4, 0, 1, 6, 13, 9, 1, 0, 1, 8, 18, 16, 6, 0, 1, 8, 24, 29, 13, 2, 0, 1, 10, 30, 43, 29, 6, 0, 1, 10, 39, 64, 52, 19, 1, 0, 1, 12, 46, 89, 89, 42, 7, 0, 1, 12, 56, 122, 139, 85, 22, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero. LINKS Alois P. Heinz, Rows n = 0..250, flattened EXAMPLE T(6,1) = 1: {[6]}. T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}. T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}. T(6,4) = 1: {[1],[2],[1,2]}. Triangle T(n,k) begins: 1; 0, 1; 0, 1; 0, 1, 2; 0, 1, 2, 1; 0, 1, 4, 2; 0, 1, 4, 6, 1; 0, 1, 6, 8, 4; 0, 1, 6, 13, 9, 1; 0, 1, 8, 18, 16, 6; 0, 1, 8, 24, 29, 13, 2; 0, 1, 10, 30, 43, 29, 6; 0, 1, 10, 39, 64, 52, 19, 1; ... MAPLE h:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1))))) end: g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add( g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n\$2), x, i), k), k=0..j)))) end: b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*g(i\$2, j), j=0..n/i)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2)): seq(T(n), n=0..14); MATHEMATICA h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]]; g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i<0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]]; b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]] ; T[n_] := CoefficientList[b[n, n], x]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give: A000007, A057427, A052928(n-1) for n>=3. Row sums give A050342. Cf. A000009, A008289, A055884, A330462, A360742, A360763. Sequence in context: A161363 A293136 A106351 * A096800 A036586 A359290 Adjacent sequences: A360761 A360762 A360763 * A360765 A360766 A360767 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Feb 19 2023 STATUS approved

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Last modified August 10 09:30 EDT 2024. Contains 375044 sequences. (Running on oeis4.)