A066183 -id:A066183 - cp's OEIS frontend

A263004 Row sums of the partition array for the products of the hook lengths numbers of Ferrers (or Young) diagrams A263003.

1, 1, 4, 15, 76, 368, 2365, 14892, 116236, 966064, 9256889, 96638496, 1129309316, 14261533248, 196315312964, 2900635720869, 45926240752560, 773725147192412, 13831256551416480, 261227089570409028, 5198858467673903360, 108706624576630569271

Offset: 0

Wolfdieter Lang, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A066183, A117506, A263003.

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, 0,
                `if`(i>n, 0, g(n-i, i, [l[], i]))+g(n, i-1, l))):
a:= n-> g(n$2, []):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 05 2015

a(n) = Sum_{k=1..A000041(n)} A263003(n,k).

a(0)=1 prepended by Alois P. Heinz, Nov 05 2015

A066184 Sum of the first moments of all partitions of n with weight starting at 1.

Original entry on oeis.org

0, 1, 5, 13, 32, 61, 123, 208, 367, 590, 957, 1459, 2266, 3328, 4938, 7097, 10205, 14299, 20100, 27626, 38023, 51485, 69600, 92882, 123863, 163235, 214798, 280141, 364530, 470660, 606557, 776233, 991370, 1258827, 1594741, 2010142, 2528445, 3165648, 3955190

Offset: 0

Wouter Meeussen, Dec 15 2001

The first element of each partition is given weight 1.

			a(3)=13 because the first moments of all partitions of 3 are {3}.{1},{2,1}.{1,2} and {1,1,1}.{1,2,3}, resulting in 3,4,6; summing to 13.

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

Cf. A066185.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      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[ Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
(* Second program: *)
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, Aug 29 2016, after Alois P. Heinz *)

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

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Emeric Deutsch, Dec 06 2015

Number of entries in row n = 1 + n^2.
Sum of entries in row n = A000041(n).
Sum(k*T(n,k), k>=0) = A066183(n).

			Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,0,0,0,1;
0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.

Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.

g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form

m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* Jean-François Alcover, Feb 19 2019 *)

G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).

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

cp's OEIS Frontend

A263004 Row sums of the partition array for the products of the hook lengths numbers of Ferrers (or Young) diagrams A263003.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

Extensions

A066184 Sum of the first moments of all partitions of n with weight starting at 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula