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A138785 Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1 <= k <= n).

%I A138785 #52 Feb 09 2021 21:46:12
%S A138785 1,2,2,4,2,3,7,6,3,4,12,8,6,4,5,19,16,12,8,5,6,30,22,18,12,10,6,7,45,
%T A138785 38,27,24,15,12,7,8,67,52,45,32,25,18,14,8,9,97,82,63,52,40,30,21,16,
%U A138785 9,10,139,112,93,72,60,42,35,24,18,10,11,195,166,135,112,85,72,49,40,27,20,11,12
%N A138785 Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1 <= k <= n).
%C A138785 T(n,k) is also the sum of all parts of size k in all partitions of n. - _Omar E. Pol_, Feb 24 2012
%C A138785 T(n,k) is also the sum of all k's that are divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - _Omar E. Pol_, Feb 05 2021
%H A138785 Alois P. Heinz, <a href="/A138785/b138785.txt">Rows n = 1..141, flattened</a>
%H A138785 R. Bacher and L. Manivel, <a href="http://www.emis.de/journals/SLC/wpapers/s47bachmani.html">Hooks and powers of parts in partitions</a>, Sem. Lotharingien de Combinatoire, 47, 2002, B47d.
%H A138785 Guo-Niu Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths</a>, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 24).
%F A138785 T(n,1) = A000070(n-1).
%F A138785 Sum_{k=1..n} k*T(n,k) = A066183(n).
%F A138785 G.f.: Sum(k*t^k*x^k/[(1-x^k)*Product(1-x^m, m=1..infinity)], k=1..infinity).
%F A138785 T(n,k) = k*A066633(n,k).
%F A138785 T(n,k) = Sum_{j=1..n} A207383(j,k). - _Omar E. Pol_, May 02 2012
%e A138785 T(4,2) = 6 because for the partitions (4), (3,1), (2,2), (2,1,1), (1,1,1,1) of n=4 the hook length multi-sets are {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1}, respectively, containing altogether six 2's.
%e A138785 Triangle starts:
%e A138785    1;
%e A138785    2,  2;
%e A138785    4,  2,  3;
%e A138785    7,  6,  3,  4;
%e A138785   12,  8,  6,  4,  5;
%e A138785   19, 16, 12,  8,  5,  6;
%e A138785   30, 22, 18, 12, 10,  6,  7;
%e A138785   45, 38, 27, 24, 15, 12,  7,  8;
%e A138785   67, 52, 45, 32, 25, 18, 14,  8, 9;
%e A138785   97, 82, 63, 52, 40, 30, 21, 16, 9, 10;
%p A138785 g:=sum(k*x^k*t^k/((1-x^k)*(product(1-x^m,m=1..50))),k=1..50): gser:= simplify(series(g,x=0,15)): for n to 12 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 12 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form
%p A138785 # second program:
%p A138785 b:= proc(n, i) option remember; `if`(n=0, [1],
%p A138785      `if`(i=1, [1, n], (p-> (g-> p(p(b(n, i-1), g),
%p A138785       [0$i, g[1]]))(`if`(i>n, [0], b(n-i, i))))(
%p A138785       (f, g)-> zip((x, y)-> x+y, f, g, 0))))
%p A138785     end:
%p A138785 T:= n-> (l-> seq(l[i+1]*i, i=1..n))(b(n$2)):
%p A138785 seq(T(n), n=1..14);  # _Alois P. Heinz_, Mar 22 2012
%t A138785 max = 12; s = Series[Sum[k*t^k*x^k/((1 - x^k)*Product[1 - x^m, {m, 1, max}]), {k, 1, max}] , {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 17 2014 *)
%t A138785 Table[Count[Flatten@IntegerPartitions@n, k]*k, {n, 12}, {k, n}] // Flatten (* _Robert Price_, Jun 15 2020 *)
%Y A138785 Row sums yield A066186.
%Y A138785 Cf. A000041, A066186, A000070, A066183, A066633.
%K A138785 nonn,tabl
%O A138785 1,2
%A A138785 _Emeric Deutsch_, May 16 2008