A180681 -id:A180681 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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

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

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

A331987 a(n) = ((n + 1) - 9(n + 1)^2 + 8(n + 1)^3)/6.

Original entry on oeis.org

0, 5, 23, 62, 130, 235, 385, 588, 852, 1185, 1595, 2090, 2678, 3367, 4165, 5080, 6120, 7293, 8607, 10070, 11690, 13475, 15433, 17572, 19900, 22425, 25155, 28098, 31262, 34655, 38285, 42160, 46288, 50677, 55335, 60270, 65490, 71003, 76817, 82940, 89380, 96145

Offset: 0

Peter Luschny, Feb 19 2020

nonn

The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.

Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Apparently a bisection of A049779 and of A024862.

Cf. A180681, A332662.

[n*(n+1)*(8*n+7)/6: n in [0..50]]; // G. C. Greubel, Apr 19 2023

a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);

LinearRecurrence[{4,-6,4,-1}, {0,5,23,62}, 42]
Table[(n-9n^2+8n^3)/6,{n,50}] (* Harvey P. Dale, Apr 11 2024 *)

def A331987(n): return n*(n+1)*(8*n+7)/6
[A331987(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).
From G. C. Greubel, Apr 19 2023: (Start)
a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).
a(n) = n*binomial(8*n+8,2)/24.
a(n) = n*(n+1)*(8*n+7)/6.
E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)

A301499 Total sum of the hook lengths over all partitions of 2n-1 having exactly n parts.

Original entry on oeis.org

1, 5, 22, 56, 139, 269, 554, 956, 1724, 2830, 4686, 7286, 11539, 17261, 26076, 38130, 55753, 79385, 113350, 158152, 220883, 303346, 415752, 562264, 759601, 1013728, 1350404, 1782342, 2346390, 3064045, 3992698, 5165042, 6666529, 8552739, 10944782, 13932362

Offset: 1

Alois P. Heinz, Mar 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..12000 (terms 1..5000 from Alois P. Heinz)

Cf. A180681.

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:
a:= n-> (p-> p[1]*(2*n-1+f(n))+p[2])(b(n-1$2)):
seq(a(n), n=1..45);

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]]]];
a[n_] := Function[p, p[[1]] (2n - 1 + f[n]) + p[[2]]][b[n - 1, n - 1]];
Array[a, 45] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

a(n) = A180681(2*n-1,n).

a(n) ~ exp(Pi*sqrt(2*n/3)) * n / (8*sqrt(3)). - Vaclav Kotesovec, May 27 2018

cp's OEIS Frontend

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

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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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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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A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.

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A301499 Total sum of the hook lengths over all partitions of 2n-1 having exactly n parts.

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Maple

Mathematica

Formula

A331987 a(n) = ((n + 1) - 9(n + 1)^2 + 8(n + 1)^3)/6.