A161680 -id:A161680 - cp's OEIS frontend

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

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

A174794 a(0) = 0 and a(n) = (4n^3 - 12n^2 + 20*n - 9)/3 for n >= 1.

Original entry on oeis.org

0, 1, 5, 17, 45, 97, 181, 305, 477, 705, 997, 1361, 1805, 2337, 2965, 3697, 4541, 5505, 6597, 7825, 9197, 10721, 12405, 14257, 16285, 18497, 20901, 23505, 26317, 29345, 32597, 36081, 39805, 43777, 48005, 52497, 57261, 62305, 67637, 73265, 79197, 85441, 92005, 98897

Offset: 0

Roger L. Bagula, Mar 29 2010

For n >= 1, a(n+1) = (4*n^3 + 8*n + 3)/3 is the number of evaluation points on the n-dimensional cube in Stenger's degree 7 cubature rule. - Franck Maminirina Ramaharo, Dec 18 2018

Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
Paul Pichler, Solving the multi-country Real Business Cycle model using a monomial rule Galerkin method, Journal of Economic Dynamics and Control Vol. 35 (2011), 240-251.
Frank Stenger, Tabulation of Certain Fully Symmetric Numerical Integration Formulas of Degree 3, 5, 7, 9, and 11, Mathematics of Computation Vol. 25 (1971), 935.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A005843, A046092, A130809, A161680.

CoefficientList[Series[x*(1 + x)*(1 + 3*x^2)/(1 - x)^4, {x, 0, 50}], x]

a[0] : 0$ a[n] := (4*n^3 - 12*n^2 + 20*n - 9)/3$ makelist(a[n], n, 0, 50); /* Martin Ettl, Oct 21 2012 */

G.f.: x*(1 + x)*(1 + 3*x^2)/(1 - x)^4.
From Franck Maminirina Ramaharo, Dec 17 2018: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 5.
a(n) = 8*binomial(n - 1, 3) + 8*binomial(n - 1, 2) + 4*binomial(n - 1, 1) + 1, n >= 1.
E.g.f.: (9 - (9 - 12*x - 4*x^3)*exp(x))/3. (End)

Definition replaced by polynomial - The Assoc. Eds. of the OEIS, Aug 10 2010

A381476 Triangle read by rows: T(n,k) is the number of subsets of {1..n} with k elements such that every pair of distinct elements has a different difference, 0 <= k <= A143824(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 2, 1, 5, 10, 6, 1, 6, 15, 14, 1, 7, 21, 26, 2, 1, 8, 28, 44, 10, 1, 9, 36, 68, 26, 1, 10, 45, 100, 60, 1, 11, 55, 140, 110, 1, 12, 66, 190, 190, 4, 1, 13, 78, 250, 304, 22, 1, 14, 91, 322, 466, 68, 1, 15, 105, 406, 676, 156

Offset: 0

Andrew Howroyd, Mar 27 2025

Equivalently, a(n) is the number of Sidon sets of {1..n} of size k.

			Triangle begins:
   0 | 1;
   1 | 1,  1;
   2 | 1,  2,  1;
   3 | 1,  3,  3;
   4 | 1,  4,  6,   2;
   5 | 1,  5, 10,   6;
   6 | 1,  6, 15,  14;
   7 | 1,  7, 21,  26,   2;
   8 | 1,  8, 28,  44,  10;
   9 | 1,  9, 36,  68,  26;
  10 | 1, 10, 45, 100,  60;
  11 | 1, 11, 55, 140, 110;
  12 | 1, 12, 66, 190, 190, 4;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..464 (rows 0..60)
Wikipedia, Sidon sequence.
Index entries for sequences related to Golomb rulers.

Columns 0..5 are A000012, A001477, A161680, A212964(n-1), A241688, A241689, A241690.
Row sums are A143823.
Cf. A003022, A143824, A325879, A382395.

row(n)={
  local(L=List());
  my(recurse(k,r,b,w)=
      if(k > n, if(r>=#L,listput(L,0)); L[1+r]++,
         self()(k+1, r, b, w);
         b+=1<

T(n,A143824(n)) = A382395(n).

A068605 Number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly two elements.

Original entry on oeis.org

2, 18, 84, 300, 930, 2646, 7112, 18360, 45990, 112530, 270204, 638820, 1490762, 3440430, 7864080, 17825520, 40107726, 89652906, 199229060, 440401500, 968883762, 2122317318, 4630511064, 10066329000, 21810380150, 47110421826, 101468601612, 217969589460, 467077692570

Offset: 2

Sharon Sela (sharonsela(AT)hotmail.com), Mar 29 2002

The sequence is the column corresponding to k=2 in A090657. - Geoffrey Critzer, Mar 06 2009

Stefano Spezia, Table of n, a(n) for n = 2..3200
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A000918, A090657, A161680.

Table[ Binomial[n, 2]*(2^n - 2), {n, 2, 30}]

def A068605(n): return n*(n-1)*((1<Chai Wah Wu, Jun 20 2025

a(n) = C(n, 2) * (2^n - 2).
O.g.f.: (4x^2/(1-2x)^3) - (2x^2/(1-x)^3). - Geoffrey Critzer, Mar 06 2009
E.g.f.: exp(x)*(2*exp(x) - 1)*x^2. - Stefano Spezia, May 06 2023

Edited and extended by Robert G. Wilson v, Apr 17 2002

A198295 Riordan array (1, x*(1+x)/(1-x^3)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 2, 3, 4, 1, 0, 0, 4, 4, 6, 5, 1, 0, 1, 2, 9, 8, 10, 6, 1, 0, 1, 3, 9, 17, 15, 15, 7, 1, 0, 0, 6, 9, 24, 30, 26, 21, 8, 1, 0, 1, 3, 18, 26, 51, 51, 42, 28, 9, 1

Offset: 0

Philippe Deléham, Jan 26 2012

Triangle T(n,k), read by rows, given by (0, 1, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonals sums: see A159284.

			Triangle begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 1, 2, 3, 4, 1
0, 0, 4, 4, 6, 5, 1
0, 1, 2, 9, 8, 10, 6, 1
0, 1, 3, 9, 17, 15, 15, 7, 1

A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Cf. Columns: A000007, A011655, A186731, A185395, A185292.

Cf. Diagonals: A000012, A001477, A161680, A000125.

Sum_{k, 0<=k<=n} T(n,k) = A001590(n+2), n>0.
Sum_{k, 0<=k<=n}T(n,k)*(-1)^(n-k) = A078056(n-1), n>0.
T(n,n) = A000012(n), T(n+1,n) = A001477(n) = n, T(n+2,n) = A161680(n) = A000217(n-1); T(n+3,n) = A000125(n-1), n>=1.
G.f.: (-1+x)*(1+x+x^2)/(-1+x^3+x*y+x^2*y). - R. J. Mathar, Aug 11 2015

A322594 a(n) = (4n^3 + 12n^2 - 4*n + 3)/3.

Original entry on oeis.org

1, 5, 25, 69, 145, 261, 425, 645, 929, 1285, 1721, 2245, 2865, 3589, 4425, 5381, 6465, 7685, 9049, 10565, 12241, 14085, 16105, 18309, 20705, 23301, 26105, 29125, 32369, 35845, 39561, 43525, 47745, 52229, 56985, 62021, 67345, 72965, 78889, 85125, 91681, 98565

Offset: 0

Franck Maminirina Ramaharo, Dec 18 2018

a(n) is the number of evaluation points on the n-dimensional cube in Lyness's degree 7 cubature rule.

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.
James Lu and David L. Darmofal, Higher-dimensional integration with gaussian weight for applications in probabilistic design, SIAM J. Sci. Comput. Vol. 26 (2004), 613-624.
James N. Lyness, Symmetric integration rules for hypercubes II. Rule projection and rule extension, Math. Comp. Vol. 19 (1965), 394-407.
James N. Lyness, Integration rules of hypercubic symmetry over a certain spherically symmetric space, Math. Comp. Vol. 19 (1965), 471-476.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A161680, A174794, A321124, A322595.

Table[(4*n^3 + 12*n^2 - 4*n + 3)/3, {n, 0, 50}]

makelist((4*n^3 + 12*n^2 - 4*n + 3)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 5.
a(n) = a(n-1) + 4*A028387(n-1), n >= 1.
a(n) = 8*binomial(n, 3) + 16*binomial(n, 2) + 4*binomial(n, 1) + 1.
G.f.: (1 + x + 11*x^2 - 5*x^3)/(1 - x)^4
E.g.f.: (1/3)*(3 + 12*x + 24*x^2 + 4*x^3)*exp(x).

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.

Original entry on oeis.org

3, 11, 21, 35, 55, 83, 121, 171, 235, 315, 413, 531, 671, 835, 1025, 1243, 1491, 1771, 2085, 2435, 2823, 3251, 3721, 4235, 4795, 5403, 6061, 6771, 7535, 8355, 9233, 10171, 11171, 12235, 13365, 14563, 15831, 17171, 18585, 20075, 21643, 23291, 25021, 26835

Offset: 0

Franck Maminirina Ramaharo, Dec 19 2018

For n >= 6, a(n) is the number of evaluating points on the hypersphere in R^n in Stoyanovas's degree 7 cubature rule.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ronald Cools, Encyclopaedia of Cubature Formulas
Ronald Cools, Monomial cubature rules since "Stroud": a compilation - part 2, Journal of Computational and Applied Mathematics - Numerical evaluation of integrals Vol. 112 (1999), 21-27.
Srebra B. Stoyanova, Cubature of the seventh degree of accuracy for the hypersphere, Journal of Computational and Applied Mathematics Vol. 84 (1997), 15-21.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: A027693.

Cf. A000292, A161680, A174794, A321124, A322594.

[(n^3 + 9*n + 14*n + 9)/3: n in [0..45]]; // Vincenzo Librandi, Jun 05 2019

Table[(n^3 + 9*n + 14*n + 9)/3, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{3,11,21,35},50] (* Harvey P. Dale, Aug 19 2020 *)

makelist((n^3 + 9*n + 14*n + 9)/3, n, 0, 50);

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.
a(n) = 2*binomial(n + 1, 3) + 6*binomial(n + 1, 2) + 2*binomial(n + 1, 1) + 1.
G.f.: (3 - x - 5*x^2 + 5*x^3)/(1 - x)^4. [Corrected by Georg Fischer, May 23 2019]
E.g.f.: (1/3)*(9 + 24*x + 12*x^2 + x^3)*exp(x).

A323294 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Offset: 0

Gus Wiseman, Jan 10 2019

nonn

			The a(4) = 11 hypergraphs:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000217, A000665, A025035, A125791, A161680, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&],Union@@#==Range[n]&]],{n,10}]

seq(n)={Vec(serlaplace(1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2))} \\ Andrew Howroyd, Aug 18 2019

a(n) = binomial(n,2) for n >= 5. - Gus Wiseman, Jan 16 2019
Binomial transform is A289837. - Gus Wiseman, Jan 16 2019
a(n) = A000217(n-1) for n >= 5. - Alois P. Heinz, Jan 24 2019
E.g.f.: 1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2. - Andrew Howroyd, Aug 18 2019

A331518 a(n) = Sum_{k=0..n} q(n,k) * !k, where q(n,k) = number of partitions of n into k distinct parts and !k = subfactorial of k.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 4, 5, 7, 10, 21, 24, 37, 49, 71, 129, 160, 227, 313, 433, 572, 1012, 1213, 1750, 2315, 3223, 4159, 5740, 8945, 11206, 15402, 20506, 27545, 36068, 48122, 61960, 94694, 116240, 158580, 205397, 276458, 352526, 470101, 596433, 781224, 1111228

Offset: 0

Ilya Gutkovskiy, Jan 19 2020

nonn

a(n) is the number of permutations of [n] whose fixed points sum to n*(n-1)/2. a(6) = 4: 143256, 231456, 312456, 523416. - Alois P. Heinz, Mar 02 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000166, A008289, A032020, A331517.

Cf. A161680, A369596.

g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 02 2024

Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] Subfactorial[k], {k, 0, n}], {n, 0, 45}]
nmax = 45; CoefficientList[Series[Sum[Subfactorial[k] x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Sum[Subfactorial[k] * x^(k*(k+1)/2) / Product[(1 - x^j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2020 *)

G.f.: Sum_{k>=0} !k * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

a(n) = A369596(n,A161680(n)). - Alois P. Heinz, Mar 02 2024

A369596 Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1, 44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1, 265, 44, 44, 53, 53, 62, 64, 29, 22, 24, 16, 16, 8, 6, 5, 4, 2, 1, 1, 0, 0, 1, 1854, 265, 265, 309, 309, 353, 362, 406, 150, 159, 126, 126, 93, 86, 44, 36, 29, 19, 19, 9, 7, 5, 4, 2, 1, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 02 2024

			T(3,0) = 2: 231, 312.
T(3,1) = 1: 132.
T(3,2) = 1: 321.
T(3,3) = 1: 213.
T(3,6) = 1: 123.
T(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Triangle T(n,k) begins:
   1;
   0, 1;
   1, 0, 0,  1;
   2, 1, 1,  1,  0,  0, 1;
   9, 2, 2,  3,  3,  2, 1, 1, 0, 0, 1;
  44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Permutation

Column k=0 gives A000166.
Column k=3 gives A000255(n-2) for n>=2.
Row sums give A000142.
Row lengths give A000124.
Reversed rows converge to A331518.
T(n,n) gives A369796.
Cf. A000217, A001710, A008289, A008290, A038720, A053632, A062869, A124327, A143946, A143947, A161680, A263753, A263756, A317527, A367955, A368338.

b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
      `if`(j=n, x^j, 1)*b(s minus {j})), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7);
# second Maple program:
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
    end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);

g[n_] := g[n] = If[n == 0, 1, n*g[n - 1] + (-1)^n];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
   If[n == 0, g[m], b[n, i-1, m] + b[n-i, Min[n-i, i-1], m-1]]];
T[n_, k_] := b[k, Min[n, k], n];
Table[Table[T[n, k], {k, 0, n*(n + 1)/2}], {n, 0, 7}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)

Sum_{k=0..A000217(n)} k * T(n,k) = A001710(n+1) for n >= 1.
Sum_{k=0..A000217(n)} (1+k) * T(n,k) = A038720(n) for n >= 1.
Sum_{k=0..A000217(n)} (n*(n+1)/2-k) * T(n,k) = A317527(n+1).
T(n,A161680(n)) = A331518(n).
T(n,A000217(n)) = 1.

cp's OEIS Frontend

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

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A174794 a(0) = 0 and a(n) = (4*n^3 - 12*n^2 + 20*n - 9)/3 for n >= 1.

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A381476 Triangle read by rows: T(n,k) is the number of subsets of {1..n} with k elements such that every pair of distinct elements has a different difference, 0 <= k <= A143824(n).

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A068605 Number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly two elements.

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A198295 Riordan array (1, x*(1+x)/(1-x^3)).

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A322594 a(n) = (4*n^3 + 12*n^2 - 4*n + 3)/3.

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A322595 a(n) = (n^3 + 9*n + 14*n + 9)/3.

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A174794 a(0) = 0 and a(n) = (4n^3 - 12n^2 + 20*n - 9)/3 for n >= 1.

A322594 a(n) = (4n^3 + 12n^2 - 4*n + 3)/3.

A322595 a(n) = (n^3 + 9n + 14n + 9)/3.