A263004 -id:A263004 - cp's OEIS frontend

A066183 Total sum of squares of parts in all partitions of n.

1, 6, 17, 44, 87, 180, 311, 558, 910, 1494, 2302, 3608, 5343, 7986, 11554, 16714, 23549, 33270, 45942, 63506, 86338, 117156, 156899, 209926, 277520, 366260, 479012, 624956, 808935, 1044994, 1340364, 1715572, 2182935, 2770942, 3499379

Offset: 1

Wouter Meeussen, Dec 15 2001

nonn

Sum of hook lengths of all boxes in the Ferrers diagrams of all partitions of n (see the Guo-Niu Han paper, p. 25, Corollary 6.5). Example: a(3) = 17 because for the partitions (3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1}, {3,2,1}, respectively; the total sum of all hook lengths is 6+5+6 = 17. - Emeric Deutsch, May 15 2008
Partial sums of A206440. - Omar E. Pol, Feb 08 2012
Column k=2 of A213191. - Alois P. Heinz, Sep 20 2013
Row sums of triangles A180681, A206561 and A299768. - Omar E. Pol, Mar 20 2018

			a(3) = 17 because the squares of all partitions of 3 are {9}, {4,1} and {1,1,1}, summing to 17.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
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], May 09 2008.

Cf. A000041, A001157, A180681, A206440, A206561, A213191, A263004, A265245, A299768.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else g:= b(n, i-1); h:= b(n-i, i);
         [g[1]+h[1], g[2]+h[2] +h[1]*i^2]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2012
# second Maple program:
g := (sum(k^2*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

Table[Apply[Plus, IntegerPartitions[n]^2, {0, 2}], {n, 30}]
(* Second program: *)
b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, g = b[n, i-1]; h = b[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + h[[1]]*i^2}]]; a[n_] :=  b[n, n][[2]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 31 2015, after Alois P. Heinz *)

a(n)=my(s); forpart(v=n,s+=sum(i=1,#v,v[i]^2));s \\ Charles R Greathouse IV, Aug 31 2015

a(n)=sum(k=1,n,sigma(k,2)*numbpart(n-k)) \\ Charles R Greathouse IV, Aug 31 2015

a(n) = Sum_{k=1..n} sigma_2(k)*numbpart(n-k), where sigma_2(k)=sum of squares of divisors of k=A001157(k). - Vladeta Jovovic, Jan 26 2002
a(n) = Sum_{k>=0} k*A265245(n,k). - Emeric Deutsch, Dec 06 2015
G.f.: g(x) = (Sum_{k>=1} k^2*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 3*sqrt(2)*Zeta(3)/Pi^3 * exp(Pi*sqrt(2*n/3)) * sqrt(n). - Vaclav Kotesovec, May 28 2018

More terms from Naohiro Nomoto, Feb 07 2002

A263003 Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 6, 24, 8, 12, 8, 24, 120, 30, 24, 20, 24, 30, 120, 720, 144, 80, 144, 72, 45, 144, 72, 80, 144, 720, 5040, 840, 360, 360, 336, 144, 240, 240, 252, 144, 360, 336, 360, 840, 5040, 40320, 5760, 2016, 1440, 2880, 1920, 630, 576, 720, 960, 1152, 448, 720, 576, 2880, 1152, 630, 1440, 1920, 2016, 5760, 40320, 362880, 45360, 13440, 7560, 8640, 12960, 3456, 2240, 4320, 3024, 2160, 8640, 6480, 1920, 1680, 1680, 2160, 4320, 5184, 1920, 3024, 2240, 8640, 6480, 3456, 7560, 12960, 13440, 45360, 362880

Offset: 0

Wolfdieter Lang, Oct 09 2015

The sequence of row lengths is A000041: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...] (partition numbers p(n)).
For the ordering of this tabf array a(n,k) see Abramowitz-Stegun (A-St) ref. pp. 831-2.
This is the array n!/A117506(n,k).
For rows 1..15 of this irregular triangle see the W. Lang link.
The row sums give A263004.
The formula given below is the one obtained from the version given, e.g., in Wybourne's book for A117506(n, k). See also the Glass-Ng reference, Theorem 1, p. 701, which gives the same formula, after rewriting using also a Vandermonde determinant.
In A. Young's third paper (Q.S.A. III, see A117506), Theorem V on p. 266, CP p. 363, f/n! (the present 1/a(n,k)) appears in the decomposition of 1 for each n, that is Sum_{k = 1..p(n)} 1/a(n,k) Sum_{j=1..d(n,k)} Y'(n,k,j) = 1, with d(n,k) = A117506(n,k), and the Young operators Y' for the standard tableaux for the k-th partition of n in A-St order.
a(n,k) also appears as normalization to obtain the idempotents NP/a(n,k). See A. Young, Q.S.A. II, p. 366, CP p. 97: NP = (1/a(n,k)) (NP)^2 for each Young tableau of the shape given by the k-th partition of n in A-St order.

			The first rows of this irregular triangle are:
n\k   1    2    3    4    5   6    7   8   9   10   11
0:    1
1:    1
2:    2    2
3:    6    3    6
4:   24    8   12    8   24
5:  120   30   24   20   24  30  120
6:  720  144   80  144   72  45  144  72  80  144  720
...
Note that the rows are in general not symmetric.
See the W. Lang link for rows n = 1..15.
a(6,6) is related to the (self-conjugate) partition (1, 2, 3) of n = 6, taken in reverse order (3, 2, 1) with the Ferrers (or Young) diagram
   _ _ _
  |_|_|_| and the hook length numbers   5  3  1 ...
  |_|_|                                 3  1
  |_|                                   1
The product gives 5*3*1*3*1*1 = 45 = a(6,6).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

Alois P. Heinz, Rows n = 0..30, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
Wolfdieter Lang, Rows 1..15.

Cf. A117506, A263004.

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)->`if`(n=0 or i=1, [h([l[], 1$n])],
               `if`(i<1, [], [g(n, i-1, l)[],
               `if`(i>n, [], g(n-i, i, [l[], i]))[]])):
T:= n-> g(n$2, [])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015

a(n,k) = Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k) - j)) with the Vandermonde determinant for the variables x_i := lambda(n,k)_i + m(n,k) - i, for i, j = 1..m(n,k), where m(n,k) is the number of parts of the k-th partition of n denoted by lambda(n,k), in the A-St order (see above). Lambda(n,k)_i stands for the i-th part of the partition lambda(n,k), sorted in nonincreasing order (this is the reverse of the A-St notation for a partition).

Row n=0 prepended by Alois P. Heinz, Nov 05 2015

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A066183 Total sum of squares of parts in all partitions of n.

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