A264052 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A259361(n)) is the number of integer partitions of n having k distinct parts occurring at least twice.
1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 6, 1, 5, 9, 1, 6, 13, 3, 8, 18, 4, 10, 23, 9, 12, 32, 12, 15, 42, 19, 1, 18, 55, 27, 1, 22, 69, 41, 3, 27, 89, 56, 4, 32, 112, 78, 9, 38, 141, 106, 12, 46, 175, 141, 23, 54, 217, 188, 31, 64, 266, 247, 49, 1, 76, 326, 321, 68, 1
Offset: 0
Examples
Triangle begins: 1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 6, 1, 5, 9, 1, 6, 13, 3, 8, 18, 4, 10, 23, 9, ... T(6,2)= 1; namely [1,1,2,2]. - _Emeric Deutsch_, Sep 19 2016
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
- FindStat - Combinatorial Statistic Finder, The number of distinct parts of a partition that occur at least twice, The number of parts from which one can subtract 2 and still get an integer partition.
- V. V. Tewari, Kronecker Coefficients For Some Near-Rectangular Partitions, arXiv:1403.5327 [math.CO], 2014, MathSciNet:3320625.
Programs
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Maple
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, x, 1), 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..25); # Alois P. Heinz, Nov 02 2015 # second Maple program: g := product((1-(1-t)*x^(2*j))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 25)): for n from 0 to 23 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, i), i = 0 .. degree(P[n])) end do; # yields sequence in triangular form - Emeric Deutsch, Nov 12 2015 -
Mathematica
T[n_, k_] := SeriesCoefficient[QPochhammer[1-t, x^2]/(t*QPochhammer[x]), {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[T[n, k], {k, 0, n}], 0], {n, 0, 25}] // Flatten (* Jean-François Alcover, Dec 11 2016 *)
Formula
From Emeric Deutsch, Nov 12 2015: (Start)
G.f.: G(t,x) = Product_{j>=1} ((1-(1-t)x^{2j})/(1-x^j)).
T(n,0) = A000009(n).
T(n,1) = A090867(n).
Sum_{k>=0} k*T(n,k) = A024786(n).
(End)
Extensions
More terms from Alois P. Heinz, Nov 02 2015
Comments