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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.

%I A264052 #49 Dec 11 2016 13:03:13
%S A264052 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,
%T A264052 19,1,18,55,27,1,22,69,41,3,27,89,56,4,32,112,78,9,38,141,106,12,46,
%U A264052 175,141,23,54,217,188,31,64,266,247,49,1,76,326,321,68,1
%N 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.
%C A264052 Row sums give A000041.
%C A264052 T(n,k) is also the number of integer partitions of n having k parts from which one can subtract 2 and still get an integer partition (mapping a partition to its conjugate sends one statistic to the other).
%C A264052 T(n,k) is also the number of integer partitions of n having k distinct even parts. Example: T(6,2)= 1, counting the partition [2,4]. - _Emeric Deutsch_, Sep 19 2016
%H A264052 Alois P. Heinz, <a href="/A264052/b264052.txt">Rows n = 0..1000, flattened</a>
%H A264052 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000257">The number of distinct parts of a partition that occur at least twice</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000256">The number of parts from which one can subtract 2 and still get an integer partition</a>.
%H A264052 V. V. Tewari, <a href="http://arxiv.org/abs/1403.5327">Kronecker Coefficients For Some Near-Rectangular Partitions</a>, arXiv:1403.5327 [math.CO], 2014, MathSciNet:3320625.
%F A264052 From _Emeric Deutsch_, Nov 12 2015: (Start)
%F A264052 G.f.: G(t,x) = Product_{j>=1} ((1-(1-t)x^{2j})/(1-x^j)).
%F A264052 T(n,0) = A000009(n).
%F A264052 T(n,1) = A090867(n).
%F A264052 Sum_{k>=0} k*T(n,k) = A024786(n).
%F A264052 (End)
%e A264052 Triangle begins:
%e A264052    1,
%e A264052    1,
%e A264052    1,  1,
%e A264052    2,  1,
%e A264052    2,  3,
%e A264052    3,  4,
%e A264052    4,  6, 1,
%e A264052    5,  9, 1,
%e A264052    6, 13, 3,
%e A264052    8, 18, 4,
%e A264052   10, 23, 9,
%e A264052   ...
%e A264052 T(6,2)= 1; namely [1,1,2,2]. - _Emeric Deutsch_, Sep 19 2016
%p A264052 b:= proc(n, i) option remember; expand(
%p A264052       `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
%p A264052       `if`(j>1, x, 1), j=0..n/i))))
%p A264052     end:
%p A264052 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A264052 seq(T(n), n=0..25);  # _Alois P. Heinz_, Nov 02 2015
%p A264052 # second Maple program:
%p A264052 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
%t A264052 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 *)
%Y A264052 Cf. A000009, A000041, A024786, A090867, A259361.
%K A264052 nonn,look,tabf
%O A264052 0,5
%A A264052 _Christian Stump_, Nov 01 2015
%E A264052 More terms from _Alois P. Heinz_, Nov 02 2015