A066183 -id:A066183 - cp's OEIS frontend

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).

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, 139, 112, 93, 72, 60, 42, 35, 24, 18, 10, 11, 195, 166, 135, 112, 85, 72, 49, 40, 27, 20, 11, 12

Offset: 1

Emeric Deutsch, May 16 2008

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

			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;

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).

Row sums yield A066186.

Cf. A000041, A066186, A000070, A066183, A066633.

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

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 *)

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

A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 6, 9, 12, 0, 1, 10, 17, 20, 20, 0, 1, 18, 39, 44, 35, 35, 0, 1, 34, 101, 122, 87, 66, 54, 0, 1, 66, 279, 392, 287, 180, 105, 86, 0, 1, 130, 797, 1370, 1119, 660, 311, 176, 128, 0, 1, 258, 2319, 5024, 4775, 2904, 1281, 558, 270, 192

Offset: 0

Alois P. Heinz, Feb 28 2013

In general, if k > 0 then column k is asymptotic to 2^((k-3)/2) * 3^(k/2) * k! * Zeta(k+1) / Pi^(k+1) * exp(Pi*sqrt(2*n/3)) * n^((k-1)/2). - Vaclav Kotesovec, May 27 2018

			Square array A(n,k) begins:
:   0,  0,   0,   0,    0,     0,     0, ...
:   1,  1,   1,   1,    1,     1,     1, ...
:   3,  4,   6,  10,   18,    34,    66, ...
:   6,  9,  17,  39,  101,   279,   797, ...
:  12, 20,  44, 122,  392,  1370,  5024, ...
:  20, 35,  87, 287, 1119,  4775, 21447, ...
:  35, 66, 180, 660, 2904, 14196, 73920, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A006128, A066186, A066183, A229325, A229326, A229327, A229328, A229329, A229330, A229331, A229332.
Rows n=0-10 give: A000004, A000012, A052548, A229354, A229355, A229356, A229357, A229358, A229359, A229360, A229361.
Main diagonal gives A252761.
Cf. A213180.

b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^k*m]))
          (b(n-p*m, p-1, k)), m=0..n/p)))
    end:
A:= (n, k)-> b(n, n, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 0, l, l + {0, First[l]*p^k*m}]][b[n - p*m, p - 1, k]], { m, 0, n/p}]]] ; a[n_, k_] := b[n, n, k][[2]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; A[n_, k_] := Sum[T[n, j]*j^k, {j, 1, n}]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 15 2016 *)

A(n,k) = Sum_{j=1..n} A066633(n,j) * j^k.

A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 20, 13, 7, 4, 35, 23, 15, 9, 5, 66, 47, 31, 19, 11, 6, 105, 75, 53, 35, 23, 13, 7, 176, 131, 93, 66, 42, 27, 15, 8, 270, 203, 151, 106, 74, 49, 31, 17, 9, 420, 323, 241, 178, 126, 86, 56, 35, 19, 10, 616, 477, 365, 272, 200, 140, 98, 63, 39, 21, 11

Offset: 1

Omar E. Pol, Feb 14 2012

From Omar E. Pol, Mar 18 2018: (Start)
In the n-th row of the triangle the first differences together with its last term give the n-th row of triangle A138785 (see below):
Row..........:  1     2       3            4               5          ...
               ---  ----   -------   ------------   ----------------
This triangle:  1;  4, 2;  9, 5, 3;  20, 13, 7, 4;  35, 23, 15, 9, 5; ...
                |   | /|   | /| /|    | / | /| /|    | / | / | /| /|
                |   |/ |   |/ |/ |    |/  |/ |/ |    |/  |/  |/ |/ |
A138785......:  1;  2, 2;  4, 2, 3;   7,  6, 3, 4;  12,  8,  6, 4, 5; ... (End)

			Triangle begins:
    1;
    4,  2;
    9,  5,  3;
   20, 13,  7,  4;
   35, 23, 15,  9,  5;
   66, 47, 31, 19, 11,  6;
  105, 75, 53, 35, 23, 13,  7;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-2 give A066186, A194552.
Right border gives A000027.
Cf. A138785, A181187.
Row sums give A066183. - Omar E. Pol, Mar 19 2018
Both A180681 and A299768 have the same row sums as this triangle. - Omar E. Pol, Mar 21 2018

Table[With[{s = IntegerPartitions[n]}, Table[Total@ Flatten@ Map[Select[#, # >= k &] &, s], {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Mar 19 2018 *)

T(n,n) = n, T(n,k) = T(n,k+1) + k * A066633(n,k) for k < n.

T(n,k) = Sum_{i=k..n} A138785(n,i).

More terms from Alois P. Heinz, Feb 14 2012

A180681 T(n,k) is the sum of the hook lengths over the partitions of n with exactly k parts.

Original entry on oeis.org

1, 3, 3, 6, 5, 6, 10, 16, 8, 10, 15, 23, 22, 12, 15, 21, 47, 44, 30, 17, 21, 28, 62, 74, 56, 40, 23, 28, 36, 104, 115, 114, 71, 52, 30, 36, 45, 130, 196, 162, 139, 89, 66, 38, 45, 55, 195, 268, 286, 227, 169, 110, 82, 47, 55, 66, 235, 395, 407, 369, 269, 204, 134, 100, 57, 66

Offset: 1

Wouter Meeussen, Sep 16 2010

Row sums equal A066183 ('Total sum of squares of parts in all partitions of n').
From Omar E. Pol, Mar 20 2018: (Start)
Both column 1 and leading diagonal give A000217, n >= 1.
Both A206561 and A299768 have the same row sums as this triangle.
Apparently the second diagonal gives A133263 without the first term. (End)

			T(5,3) = 22 since the partitions of 5 in 3 parts are 221 and 311, with hook lengths {{2,4}, {1,3}, {1}} and {{1,2,5}, {2}, {1}} summing to 22.
Triangle T(n,k) begins:
   1;
   3,   3;
   6,   5,   6;
  10,  16,   8,  10;
  15,  23,  22,  12,  15;
  21,  47,  44,  30,  17,  21;
  28,  62,  74,  56,  40,  23,  28;
  36, 104, 115, 114,  71,  52,  30, 36;
  45, 130, 196, 162, 139,  89,  66, 38, 45;
  55, 195, 268, 286, 227, 169, 110, 82, 47, 55;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000217, A008284, A066183, A133263, A206561, A299768.

T(2n-1,n) gives A301499.

f:= n-> (n-1)*n/2:
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n+f(n)],
      b(n, i-1)+(p-> p+[0, p[1]*(n+f(i))])(b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> (p-> p[1]*(n+f(k))+p[2])(b(n-k, min(n-k, k))):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Mar 20 2018

(*Needs["DiscreteMath`Combinatorica`"]; hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[ #1], Max[p]] & ) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] & ) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] & ) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, {2}] - 1]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); Table[Tr[ Tr[ Flatten[hooklength[ # ]]] &/@ partitionexact[n,k] ] ,{n,16},{k,n}]
(* Second program: *)
Table[p = IntegerPartitions[n, {k}]; Total@Table[y = Table[Boole[p[[l]][[i]] >= j], {i, k}, {j, n}]; Total[Table[Total[{y[[i, j ;; n]], y[[i + 1 ;; k, j]]}, 2], {i, k}, {j, n}], 2], {l, Length[p]}], {n, 11}, {k, n}] // Flatten (* Robert Price, Jun 19 2020 *)
f[n_] := n(n-1)/2;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n + f[n]}, b[n, i - 1] + Function[p, p + {0, p[[1]] (n + f[i])}][b[n - i, Min[n - i, i]]]];
T[n_, k_] := Function[p, p[[1]] (n + f[k]) + p[[2]]][b[n-k, Min[n-k, k]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

A066185 Sum of the first moments of all partitions of n with weights starting at 0.

Original entry on oeis.org

0, 0, 1, 4, 12, 26, 57, 103, 191, 320, 537, 843, 1342, 2015, 3048, 4457, 6509, 9250, 13170, 18316, 25483, 34853, 47556, 64017, 86063, 114285, 151462, 198871, 260426, 338275, 438437, 564131, 724202, 924108, 1176201, 1489237, 1881273, 2365079, 2966620, 3705799

Offset: 0

Wouter Meeussen, Dec 15 2001

The first element of each partition is given weight 0.
Consider the partitions of n, e.g., n=5. For each partition sum T(e-1) and sum all these. E.g., 5 -> T(4)=10, 41 -> T(3)+T(0)=6, 32 -> T(2)+T(1)=4, 311 -> T(2)+T(0)+T(0)=3, 221 -> T(1)+T(1)+T(0)=2, 21111 ->1 and 11111 ->0. Summing, 10+6+4+3+2+1+0 = 26 as desired. - Jon Perry, Dec 12 2003
Also equals the sum of f(p) over the partitions p of n, where f(p) is obtained by replacing each part p_i of partition p by p_i*(p_i-1)/2. See I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition, p. 3, eqn (1.5) and (1.6). - Wouter Meeussen, Sep 25 2014

			a(3)=4 because the first moments of all partitions of 3 are {3}.{0},{2,1}.{0,1} and {1,1,1}.{0,1,2}, resulting in 0,1,3; summing to 4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000337, A001788, A066183, A066184, A066186, A161680, A264034.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0],
      b(n, i-1)+(h-> h+[0, h[1]*i*(i-1)/2])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

Table[ Plus@@ Map[ #.Range[ 0, -1+Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], b[n, i-1] + Function[h, h+{0, h[[1]]*i*(i-1)/2}][b[n-i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

a(n) = 1/2*(A066183(n) - A066186(n)). - Vladeta Jovovic, Mar 23 2003
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021
a(n) = Sum_{k=0..A161680(n)} k * A264034(n,k). - Alois P. Heinz, Jan 20 2023
a(n) ~ 3 * zeta(3) * sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2) * Pi^3). - Vaclav Kotesovec, Jul 06 2025

A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

Original entry on oeis.org

1, 5, 11, 27, 43, 93, 131, 247, 352, 584, 808, 1306, 1735, 2643, 3568, 5160, 6835, 9721, 12672, 17564, 22832, 30818, 39743, 53027, 67594, 88740, 112752, 145944, 183979, 236059, 295370, 375208, 467363, 588007, 728437, 910339, 1121009, 1391083, 1706003, 2103013

Offset: 1

Omar E. Pol, Feb 08 2012

nonn

Since partial sums of this sequence give A066183 we have that A066183(n) is also the volume of the mentioned version of the shell model of partitions with n shells. Each part of size k has a volume equal to k^2 since each box is a cuboid whose sides have lengths: 1, k, k.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Row sums of triangle A206438. Partial sums give A066183.

Cf. A135010, A138121, A141285.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p-> p+[0, p[1]*i^2])(b(n-i, min(n-i, i))))
    end:
a:= n-> (b(n$2)-b(n-1$2))[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2022

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n},
     b[n, i-1] + Function[p, p + {0, p[[1]]*i^2}][b[n-i, Min[n-i, i]]]];
a[n_] := (b[n, n] - b[n-1, n-1])[[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)

a(n) ~ sqrt(3) * zeta(3) * exp(Pi*sqrt(2*n/3)) / Pi^2. - Vaclav Kotesovec, Oct 20 2024

A299768 Triangle read by rows: T(n,k) = sum of all squares of the parts k in all partitions of n, with n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 4, 4, 4, 9, 7, 12, 9, 16, 12, 16, 18, 16, 25, 19, 32, 36, 32, 25, 36, 30, 44, 54, 48, 50, 36, 49, 45, 76, 81, 96, 75, 72, 49, 64, 67, 104, 135, 128, 125, 108, 98, 64, 81, 97, 164, 189, 208, 200, 180, 147, 128, 81, 100, 139, 224, 279, 288, 300, 252, 245, 192, 162, 100, 121

Offset: 1

Omar E. Pol, Mar 19 2018

			Triangle begins:
   1;
   2,  4;
   4,  4,  9;
   7, 12,  9, 16;
  12, 16, 18, 16, 25,
  19, 32, 36, 32, 25, 36;
  30, 44, 54, 48, 50, 36, 49;
...
For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1], so the squares of the parts are respectively [16], [4, 4], [9, 1], [4, 1, 1], [1, 1, 1, 1]. The sum of the squares of the parts 1 is 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7. The sum of the squares of the parts 2 is 4 + 4 + 4 = 12. The sum of the squares of the parts 3 is 9. The sum of the squares of the parts 4 is 16. So the fourth row of triangle is [7, 12, 9, 16].

Alois P. Heinz, Rows n = 1..200, flattened

Column 1 is A000070.
Leading diagonal is A000290, n >= 1.
Row sums give A066183.
Both A180681 and A206561 have the same row sums as this triangle.
Cf. A066633, A138785.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
      (p-> p+(coeff(p, x, 0)*i^2)*x^i)(b(n-i, min(n-i, i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 20 2018

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1 + n*x, b[n, i - 1] + # + (Coefficient[#, x, 0]*i^2*x^i)&[b[n - i, Min[n - i, i]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, n}]&[b[n, n]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, May 22 2018, after Alois P. Heinz *)

row(n) = {v = vector(n); forpart(p=n, for(k=1, #p, v[p[k]] += p[k]^2;);); v;} \\ Michel Marcus, Mar 20 2018

T(n,k) = (k^2)*A066633(n,k) = k*A138785(n,k). - Omar E. Pol, Jun 07 2018

More terms from Michel Marcus, Mar 20 2018

A027992 a(n) = 1T(n,0) + 2T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926.

Original entry on oeis.org

1, 6, 22, 66, 178, 450, 1090, 2562, 5890, 13314, 29698, 65538, 143362, 311298, 671746, 1441794, 3080194, 6553602, 13893634, 29360130, 61865986, 130023426, 272629762, 570425346, 1191182338, 2483027970, 5167382530, 10737418242

Offset: 0

Clark Kimberling

nonn

Also total sum of squares of parts in all compositions of n (offset 1). Total sum of cubes of parts in all compositions of n is (13*n-36)*2^(n-1)+6*n+18 with g.f. x*(1+4x+x^2)/((2x-1)(1-x))^2, A271638; total sum of fourth powers of parts in all compositions of n is (75*n-316)*2^(n-1)+12*n^2+72*n+158 with g.f. x*(1+x)*(x^2+10*x+1)/((2*x-1)^2*(1-x)^3); total sum of fifth powers of parts in all compositions of n is (541*n-3060)*2^(n-1)+20*n^3+180*n^2+790*n+1530. - Vladeta Jovovic, Mar 18 2005

Let M = the 3 X 3 matrix [(1,0,0),(1,2,0),(1,3,2)] and column vector V = [1,1,1]. a(n) is the lower term in the product M^n * V.

Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
Alejandro Erickson and Mark Schurch, Enumerating tatami mat arrangements of square grids, in 22nd International Workshop on Combinatorial Algorithms, University of Victoria, June 20-22, volume 7056 of Lecture Notes in Computer Science (LNCS), Springer Berlin / Heidelberg, 2011, pp. 223-235
K. Kimura, S. Higuchi, Monte Carlo estimation of the number of tatami tilings, arXiv:1509.05983 [cond-mat.stat-mech], 2015-2016, eq. (2).

Cf. A027926, A066183.

M = {{1, 0, 0}, {1, 2, 0}, {1, 3, 2}};
a[n_] := MatrixPower[M, n].{1, 1, 1} // Last;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 12 2018, from PARI *)

vector(40, n, n--; ([1,0,0;1,2,0;1,3,2]^n*[1,1,1]~)[3]) \\ Michel Marcus, Aug 06 2015

a(n) = 2^n*(3n-1)+2 = A048496(n+1)-1 = A053565(n+1)+2. - Ralf Stephan, Jan 15 2004

a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). G.f.: (1+x)/((1-x)*(1-2*x)^2). - Colin Barker, Apr 04 2012

A206438 Triangle read by rows which lists the squares of the parts of A135010.

Original entry on oeis.org

1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 4, 16, 1, 1, 1, 1, 1, 4, 9, 25, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 16, 9, 9, 36, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 9, 4, 25, 9, 16, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 16, 4, 9, 9, 4, 36, 9, 25

Offset: 1

Omar E. Pol, Feb 08 2012

Volumes of the parts in the section model of partitions version "boxes" in which each part of size k has a volume = k^2. Row sums of this triangle give A206440 and partial sums of A206440 give A066183.

			Written as a triangle:
1;
1,4;
1,1,9;
1,1,1,4,4,16;
1,1,1,1,1,4,9,25;
1,1,1,1,1,1,1,4,4,4,4,16,9,9,36;
1,1,1,1,1,1,1,1,1,1,1,4,4,9,4,25,9,16,49;

Alois P. Heinz, Rows n = 1..23, flattened

Row n has length A138137(n).
Row sums give A206440.
Right border gives positives A000290.
Cf. A066183, A135010, A138121, A141285.

Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 8}] ^2  // Flatten (* Robert Price, May 28 2020 *)

a(n) = A135010(n)^2.

A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

Original entry on oeis.org

0, 1, 3, 7, 16, 28, 59, 91, 170, 269, 450, 655, 1162, 1602, 2527, 3793, 5805, 8034, 12660, 17131, 26484, 37384, 53738, 73504, 114683, 153613, 221225, 313339, 453769, 609179, 927968, 1223909, 1804710, 2522264, 3539835, 4855420, 7439870, 9765555, 14009545

Offset: 0

Alois P. Heinz, Feb 27 2013

nonn

			a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);

b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

From Vaclav Kotesovec, May 24 2018: (Start)
a(n) ~ c * 3^(n/3), where
c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0
c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1
c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2
(End)

cp's OEIS Frontend

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).

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A213191 Total sum A(n,k) of k-th powers of parts in all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

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A180681 T(n,k) is the sum of the hook lengths over the partitions of n with exactly k parts.

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A066185 Sum of the first moments of all partitions of n with weights starting at 0.

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A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

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A299768 Triangle read by rows: T(n,k) = sum of all squares of the parts k in all partitions of n, with n >= 1, 1 <= k <= n.

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A027992 a(n) = 1T(n,0) + 2T(n,1) + ... + (2n+1)*T(n,2n), T given by A027926.