A206440 -id:A206440 - 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

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.

A299769 Triangle read by rows: T(n,k) is the sum of all squares of the parts k in the last section of the set of partitions of n, with n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 1, 4, 2, 0, 9, 3, 8, 0, 16, 5, 4, 9, 0, 25, 7, 16, 18, 16, 0, 36, 11, 12, 18, 16, 25, 0, 49, 15, 32, 27, 48, 25, 36, 0, 64, 22, 28, 54, 32, 50, 36, 49, 0, 81, 30, 60, 54, 80, 75, 72, 49, 64, 0, 100, 42, 60, 90, 80, 100, 72, 98, 64, 81, 0, 121, 56, 108, 126, 160, 125, 180, 98, 128, 81, 100, 0, 144

Offset: 1

Omar E. Pol, Mar 20 2018

The partial sums of the k-th column of this triangle give the k-th column of triangle A299768.

Note that the last section of the set of partitions of n is also the n-th section of the set of partitions of any positive integer >= n.

			Triangle begins:
   1;
   1,   4;
   2,   0,   9;
   3,   8,   0,  16;
   5,   4,   9,   0,  25;
   7,  16,  18,  16,   0,  36;
  11,  12,  18,  16,  25,   0,  49;
  15,  32,  27,  48,  25,  36,   0,  64;
  22,  28,  54,  32,  50,  36,  49,   0,  81;
  30,  60,  54,  80,  75,  72,  49,  64,   0, 100;
  42,  60,  90,  80, 100,  72,  98,  64,  81,   0, 121;
  56, 108, 126, 160, 125, 180,  98, 128,  81, 100,   0, 144;
  ...
Illustration for the 4th row of triangle:
.
.                                  Last section of the set
.        Partitions of 4.          of the partitions of 4.
.       _ _ _ _                              _
.      |_| | | |  [1,1,1,1]                 | |  [1]
.      |_ _| | |  [2,1,1]                   | |  [1]
.      |_ _ _| |  [3,1]                _ _ _| |  [1]
.      |_ _|   |  [2,2]               |_ _|   |  [2,2]
.      |_ _ _ _|  [4]                 |_ _ _ _|  [4]
.
For n = 4 the last section of the set of partitions of 4 is [4], [2, 2], [1], [1], [1], so the squares of the parts are respectively [16], [4, 4], [1], [1], [1]. The sum of the squares of the parts 1 is 1 + 1 + 1 = 3. The sum of the squares of the parts 2 is 4 + 4 = 8. The sum of the squares of the parts 3 is 0 because there are no parts 3. The sum of the squares of the parts 4 is 16. So the fourth row of triangle is [3, 8, 0, 16].

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

Column 1 is A000041.
Leading diagonal gives A000290, n >= 1.
Second diagonal gives A000007.
Row sums give A206440.
Cf. A066183, A135010, A206438, A299768.

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)-b(n-1$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Jul 23 2018

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1 + n*x, b[n, i-1] + Function[p, p + (Coefficient[p, x, 0]*i^2)*x^i][b[n-i, Min[n-i, i]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n] - b[n-1, n-1]];
T /@ Range[14] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P.heinz *)

T(n,k) = A299768(n,k) - A299768(n-1,k). - Alois P. Heinz, Jul 23 2018

A299772 Triangle read by rows T(n,k) in which the partial sums of column k give the column k of triangle A180681.

Original entry on oeis.org

1, 2, 3, 3, 2, 6, 4, 11, 2, 10, 5, 7, 14, 2, 15, 6, 24, 22, 18, 2, 21, 7, 15, 30, 26, 23, 2, 28, 8, 42, 41, 58, 31, 29, 2, 36, 9, 26, 81, 48, 68, 37, 36, 2, 45, 10, 65, 72, 124, 88, 80, 44, 44, 2, 55, 11, 40, 127, 121, 142, 100, 94, 52, 53, 2, 66, 12, 93, 156, 232, 177, 208, 114, 110, 61, 63, 2, 78

Offset: 1

Omar E. Pol, Mar 20 2018

			Triangle begins:
   1;
   2,  3;
   3,  2,   6;
   4, 11,   2,  10;
   5,  7,  14,   2,  15;
   6, 24,  22,  18,   2,  21;
   7, 15,  30,  26,  23,   2,  28;
   8, 42,  41,  58,  31,  29,   2,  36;
   9, 26,  81,  48,  68,  37,  36,   2, 45;
  10, 65,  72, 124,  88,  80,  44,  44,  2, 55;
  11, 40, 127, 121, 142, 100,  94,  52, 53,  2, 66;
  12, 93, 156, 232, 177, 208, 114, 110, 61, 63,  2, 78;
...

Column 1 is A000027.
Leading diagonal is A000217.
Row sums give A206440.
Apparently the second diagonal gives A007395.
Cf. A180681.

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

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

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A299769 Triangle read by rows: T(n,k) is the sum of all squares of the parts k in the last section of the set of partitions of n, with n >= 1, 1 <= k <= n.

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A299772 Triangle read by rows T(n,k) in which the partial sums of column k give the column k of triangle A180681.

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