A001117 -id:A001117 - cp's OEIS frontend

A363591 a(n) = 3*(3^(n-1) - 2^n + 1)/2 - binomial(n,2), n >= 3.

0, 12, 65, 255, 882, 2870, 9039, 27945, 85448, 259512, 784797, 2366819, 7125198, 21424938, 64373339, 193316877, 580344132, 1741819148, 5227030665, 15684238119, 47059006250, 141189602142, 423593972775, 1270832250545, 3812597415552, 11437993573920, 34314383375669

Offset: 3

Enrique Navarrete, Jun 10 2023

2*a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the number of elements in the first two sets (in total) is at least three.

			2*a(5)=130 subtracting the 20 ordered set partitions of the type {1},{2},{3,4,5} from the 150 ordered set partitions of a 5-set into 3 parts.

Paolo Xausa, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,34,-23,6).

Cf. A001117, A363603.

LinearRecurrence[{8, -24, 34, -23, 6}, {0, 12, 65, 255, 882}, 30] (* or *)
A363591[n_] := (3^n - 3*2^n - n^2 + n + 3)/2;
Array[A363591, 30, 3] (* Paolo Xausa, Aug 30 2024 *)

G.f.: x^4*(12 - 31*x + 23*x^2 - 6*x^3)/((1 - x)^3*(1 - 2*x)*(1 - 3*x)). - Stefano Spezia, Jun 11 2023

A363603 Expansion of e.g.f. (1/4)(exp(x)-x-1)(exp(x)-1)^2.

Original entry on oeis.org

3, 20, 90, 343, 1197, 3966, 12720, 39941, 123651, 379132, 1154790, 3501219, 10581465, 31908218, 96068700, 288926977, 868288239, 2608010424, 7830584850, 23505386015, 70544469573, 211692128950, 635198021640, 1905845723133, 5718057263067

Offset: 4

Enrique Navarrete, Jun 11 2023

4*a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the number of elements in a particular set (say the first one) is at least two (see example).

4*a(n) is also the number of ternary strings using digits {0,1,2} so that all digits are used and a particular digit appears at least twice; for example, for n=5, the 80 strings with at least two 0's are 00112 (30 of this type), 00122 (30 of this type), 00012 (20 of this type).

			4*a(5)=80 since the ordered set partitions are the following: 30 of type {1,2}{3,4},{5}; 30 of type {1,2},{3},{4,5}; 20 of type {1,2,3},{4},{5}.

Paolo Xausa, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12).

Cf. A000225, A052749, A001117, A363591.

A363603[n_]:=(3^n-3(2^n-1))/4-(n/2)(2^(n-2)-1);Array[A363603,40,4] (* or *)
LinearRecurrence[{9,-31,51,-40,12},{3,20,90,343,1197},40] (* Paolo Xausa, Nov 18 2023 *)

a(n) = (3^n - 3*(2^n - 1))/4 - (n/2)*(2^(n-2) - 1), n>=4.
G.f.: x^4*(3 - 7*x + 3*x^2)/((1 - 3*x)*(1 - 2*x)^2*(1 - x)^2). - Stefano Spezia, Jun 11 2023
a(n) = (Sum_{k=2..n-2} A000225(k-1)*binomial(n,k))/2. - R. J. Cano, Jul 27 2023

A371568 Array read by ascending antidiagonals: A(n, k) is the number of paths of length k in Z^n from the origin to points such that x1+x2+...+xn = k with x1,...,xn > 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 14, 1, 0, 0, 0, 36, 30, 1, 0, 0, 0, 24, 150, 62, 1, 0, 0, 0, 0, 240, 540, 126, 1, 0, 0, 0, 0, 120, 1560, 1806, 254, 1, 0, 0, 0, 0, 1800, 8400, 5796, 510, 1

Offset: 1

Shel Kaphan, Mar 28 2024

T(n, k) can also be seen as the number of ordered partitions of k items into n nonempty buckets.
T(n, n) = n!, which is readily seen because to go from the origin to a point in Z^n a distance n away, with at least one step taken in each dimension, the first step can be in any of n dimensions, the second step in any of n-1 dimensions, and so on.
This array is the image of Pascal's triangle A007318 under the Akiyama-Tanigawa transformation. See the Python program. - Peter Luschny, Apr 19 2024

			 n\k 1 2 3  4   5    6     7      8       9       10
  --------------------------------------------------
 1|  1 1 1  1   1    1     1      1       1        1
 2|  0 2 6 14  30   62   126    254     510     1022
 3|  0 0 6 36 150  540  1806   5796   18150    55980
 4|  0 0 0 24 240 1560  8400  40824  186480   818520
 5|  0 0 0  0 120 1800 16800 126000  834120  5103000
 6|  0 0 0  0   0  720 15120 191520 1905120 16435440
 7|  0 0 0  0   0    0  5040 141120 2328480 29635200
 8|  0 0 0  0   0    0     0  40320 1451520 30240000
 9|  0 0 0  0   0    0     0      0  362880 16329600
10|  0 0 0  0   0    0     0      0       0  3628800

Cf. A000918 (n=2), A001117 (n=3), A000919 (n=4), A001118 (n=5), A000920 (n=6).
Cf. A135456 (n=7), A133068 (n=8), A133360 (n=9), A133132 (n=10).
Cf. A371064, A007318.
See A019538 and A131689 for other versions.

A[n_,k_] := Sum[(-1)^(n-i) * i^k * Binomial[n,i], {i,1,n}]

# The Akiyama-Tanigawa algorithm for the binomial generates the rows.
# Adds row(0) = 0^k and column(0) = 0^n.
from math import comb as binomial
def ATBinomial(n, len):
    A = [0] * len
    R = [0] * len
    for k in range(len):
        R[k] = binomial(k, n)
        for j in range(k, 0, -1):
            R[j - 1] = j * (R[j] - R[j - 1])
        A[k] = R[0]
    return A
for n in range(11): print([n], ATBinomial(n, 11))  # Peter Luschny, Apr 19 2024

A(n,k) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * i^k

A056268 Number of primitive (aperiodic) words of length n which contain exactly three different symbols.

Original entry on oeis.org

0, 0, 6, 36, 150, 534, 1806, 5760, 18144, 55830, 171006, 518580, 1569750, 4732014, 14250450, 42844320, 128746950, 386615376, 1160688606, 3483582660, 10454059938, 31368305694, 94118013006, 282378679920, 847187946000, 2541662931990, 7625194813656, 22875982414740

Offset: 1

Marks R. Nester

nonn

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A000244, A001117, A008683, A054718.

f(n) = 3^n - 3*2^n + 3;
a(n) = sumdiv(n, d, moebius(d)*f(n/d)); \\ Michel Marcus, Mar 25 2022

a(n) = Sum_{d|n} mu(d)*A001117(n/d).

a(11) and following corrected by Georg Fischer, Mar 24 2022

A179483 A(k,3) where A(k,n) = Sum_{m=1..k} (-1)^(m+1) binomial(n,m)m^k.

Original entry on oeis.org

3, -9, 6, 36, 150, 540, 1806, 5796, 18150, 55980, 171006, 519156, 1569750, 4733820, 14250606, 42850116, 128746950, 386634060, 1160688606, 3483638676, 10454061750, 31368476700, 94118013006, 282379204836, 847187946150, 2541664501740, 7625194831806

Offset: 1

M. Lawrence Glasser, Jul 16 2010

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A001117.

A179483 := proc(n) add( (-1)^(m+1)*binomial(3,m)*m^n,m=1..n) ; end proc: # R. J. Mathar, Jan 31 2011

Mathematica
```
Sum[(-1)^(m+1)Binomial[3,m]m^k,{m,1,k}]
```

Vec(3*x*(1 - 9*x + 31*x^2 - 39*x^3 + 18*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 21 2017

a(n) = A001117(n), n>=3. - R. J. Mathar, Jul 20 2010
From Colin Barker, May 21 2017: (Start)
G.f.: 3*x*(1 - 9*x + 31*x^2 - 39*x^3 + 18*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 3 - 3*2^n + 3^n for n>2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5.
(End)

A380993 Irregular triangular array read by rows. T(n,k) is the number of ternary words of length n containing at least one copy of each letter and having exactly k inversions, n>=3, 0<=k<=floor(n^2/3).

Original entry on oeis.org

1, 2, 2, 1, 3, 6, 9, 9, 6, 3, 6, 12, 21, 27, 30, 24, 18, 9, 3, 10, 20, 38, 55, 74, 81, 80, 69, 53, 34, 17, 8, 1, 15, 30, 60, 93, 138, 174, 210, 216, 219, 195, 165, 120, 84, 48, 27, 9, 3, 21, 42, 87, 141, 222, 303, 405, 480, 546, 579, 588, 552, 498, 414, 324, 240, 162, 99, 54, 27, 9, 3

Offset: 3

Geoffrey Critzer, Feb 11 2025

			Triangle T(n,k) begins:
   1,  2,  2,  1;
   3,  6,  9,  9,  6,  3;
   6, 12, 21, 27, 30, 24, 18,  9,  3;
  10, 20, 38, 55, 74, 81, 80, 69, 53, 34, 17, 8, 1;
  ...
T(4,2) = 9 because we have: {0, 1, 2, 0}, {0, 2, 0, 1}, {0, 2, 1, 1}, {0, 2, 2, 1}, {1, 0, 0, 2}, {1, 0, 2, 1}, {1, 1, 0, 2}, {1, 2, 0, 2}, {2, 0, 1, 2}.

Alois P. Heinz, Rows n = 3..50, flattened

Cf. A056454, A129529, A001117 (row sums).

b:= proc(n, l) option remember; `if`(n=0, `if`(nops(subs(0=
      [][], l))=3, 1, 0), add(expand(x^([0, l[1], l[1]+l[2]][j])*
      b(n-1, subsop(j=`if`(j=3, 1, l[j]+1), l))), j=1..3))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
seq(T(n), n=3..10);  # Alois P. Heinz, Feb 12 2025

nn = 8; B[n_] := FunctionExpand[QFactorial[n, u]];
e[z_] := Sum[z^n/B[n], {n, 0, nn}];
Drop[Map[CoefficientList[#, u] &,
   Map[Normal[Series[#, {u, 0, Binomial[nn, 2]}]] &,
    Table[B[n], {n, 0, nn}] CoefficientList[
      Series[(e[z] - 1)^3, {z, 0, nn}], z]]], 3] // Grid

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/B(n) = (e(x)-1)^3 where B(n) = Product_{i=1..n} (q^i-1)/(q-1) and e(x) = Sum_{n>=0} x^n/B(n).

Sum_{k=0..floor(n^2/3)} (-1)^k * T(n,k) = A056454(n). - Alois P. Heinz, Feb 12 2025

cp's OEIS Frontend

A363591 a(n) = 3*(3^(n-1) - 2^n + 1)/2 - binomial(n,2), n >= 3.

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A363603 Expansion of e.g.f. (1/4)*(exp(x)-x-1)*(exp(x)-1)^2.

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A371568 Array read by ascending antidiagonals: A(n, k) is the number of paths of length k in Z^n from the origin to points such that x1+x2+...+xn = k with x1,...,xn > 0.

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A056268 Number of primitive (aperiodic) words of length n which contain exactly three different symbols.

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A179483 A(k,3) where A(k,n) = Sum_{m=1..k} (-1)^(m+1) *binomial(n,m)*m^k.

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A380993 Irregular triangular array read by rows. T(n,k) is the number of ternary words of length n containing at least one copy of each letter and having exactly k inversions, n>=3, 0<=k<=floor(n^2/3).

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Author

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Examples

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Programs

Maple

Mathematica

Formula

A363603 Expansion of e.g.f. (1/4)(exp(x)-x-1)(exp(x)-1)^2.

A179483 A(k,3) where A(k,n) = Sum_{m=1..k} (-1)^(m+1) binomial(n,m)m^k.