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 A276424 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once. 4
 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 4, 0, 2, 0, 1, 0, 6, 0, 2, 0, 1, 0, 2, 8, 0, 3, 0, 2, 0, 2, 0, 11, 0, 4, 0, 3, 0, 2, 0, 2, 15, 0, 5, 0, 4, 0, 4, 0, 2, 0, 19, 0, 7, 0, 6, 0, 5, 0, 2, 0, 3, 25, 0, 9, 0, 8, 0, 7, 0, 4, 0, 3, 0, 34, 0, 11, 0, 10, 0, 10, 0, 5, 0, 3, 0, 4 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS T(n,0) = A265254(n). T(n,n) = A035457(n). Sum_{k>=0} k*T(n,k) = A276425(n). Sum_{k>=0} T(n,k) = A000041(n). LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA G.f.: G(t,x) = Product_{j>=1}(((1-x^{2j})(1+t^{2j}x^{2j}) + x^{4j})/(1-x^j). EXAMPLE Row 4 is 3, 0, 1, 0, 1 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively. Row 5 is 4, 0, 2, 0, 1, 0 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the even singletons are 0, 2, 0, 0, 2, 4, 0, respectively. Triangle starts: 1; 1,0; 1,0,1; 2,0,1,0; 3,0,1,0,1; 4,0,2,0,1,0; 6,0,2,0,1,0,2. MAPLE g := Product(((1-x^(2*j))*(1+t^(2*j)*x^(2*j))+x^(4*j))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, i), i = 0 .. n) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, i) option remember; expand(       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*       `if`(j=1 and i::even, x^i, 1), j=0..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n\$2)): seq(T(n), n=0..14);  # Alois P. Heinz, Sep 14 2016 MATHEMATICA b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && EvenQ[i], x^i, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2016, after Alois P. Heinz *) CROSSREFS Cf. A000041, A035457, A265254, A276422, A276423, A276425. Sequence in context: A093057 A065334 A162590 * A191258 A254990 A191255 Adjacent sequences:  A276421 A276422 A276423 * A276425 A276426 A276427 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Sep 14 2016 STATUS approved

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Last modified April 16 00:05 EDT 2021. Contains 343020 sequences. (Running on oeis4.)