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A268193
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Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.
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3
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1, 2, 2, 1, 4, 1, 4, 3, 8, 2, 1, 8, 6, 1, 13, 7, 2, 15, 11, 4, 22, 15, 4, 1, 24, 24, 7, 1, 37, 26, 12, 2, 40, 42, 16, 3, 57, 50, 22, 6, 64, 72, 33, 6, 1, 89, 84, 46, 11, 1, 98, 122, 60, 15, 2, 135, 141, 82, 24, 3, 149, 198, 106, 32, 5, 199, 231, 144, 45, 8, 224, 309, 187, 61, 10, 1
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OFFSET
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1,2
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COMMENTS
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T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
Sum of entries in row n is A000041(n) (the partition numbers).
Sum(k*T(n,k), k>=1) = A024786(n-1).
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LINKS
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FORMULA
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G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).
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EXAMPLE
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T(5,1) = 3 because we have [3,2], [[2,2,1], and [2,1,1,1].
T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
Triangle starts:
1;
2;
2,1;
4,1;
4,3;
8,2,1;
8,6,1;
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MAPLE
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g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
`if`(t and j=1, x, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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