OFFSET
1,2
COMMENTS
T(n,k) is also the sum of all parts of size k in all partitions of n. - Omar E. Pol, Feb 24 2012
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
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
R. Bacher and L. Manivel, Hooks and powers of parts in partitions, Sem. Lotharingien de Combinatoire, 47, 2002, B47d.
Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 24).
FORMULA
T(n,1) = A000070(n-1).
Sum_{k=1..n} k*T(n,k) = A066183(n).
G.f.: Sum(k*t^k*x^k/[(1-x^k)*Product(1-x^m, m=1..infinity)], k=1..infinity).
T(n,k) = k*A066633(n,k).
T(n,k) = Sum_{j=1..n} A207383(j,k). - Omar E. Pol, May 02 2012
EXAMPLE
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.
Triangle starts:
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, 38, 27, 24, 15, 12, 7, 8;
67, 52, 45, 32, 25, 18, 14, 8, 9;
97, 82, 63, 52, 40, 30, 21, 16, 9, 10;
MAPLE
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
# second program:
b:= proc(n, i) option remember; `if`(n=0, [1],
`if`(i=1, [1, n], (p-> (g-> p(p(b(n, i-1), g),
[0$i, g[1]]))(`if`(i>n, [0], b(n-i, i))))(
(f, g)-> zip((x, y)-> x+y, f, g, 0))))
end:
T:= n-> (l-> seq(l[i+1]*i, i=1..n))(b(n$2)):
seq(T(n), n=1..14); # Alois P. Heinz, Mar 22 2012
MATHEMATICA
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 *)
Table[Count[Flatten@IntegerPartitions@n, k]*k, {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, May 16 2008
STATUS
approved