A293181 -id:A293181 - cp's OEIS frontend

A304975 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets).

0, 0, 0, 0, 0, 1, 3, 12, 34, 95, 261, 630, 1700, 3801, 10143, 21672, 57414, 119155, 314121, 639210, 1679320, 3370301, 8832483, 17549532, 45907994, 90541815, 236526381, 463889790, 1210585740, 2364180001, 6164760423, 11999840592, 31271161774, 60714998075, 158145313041, 306438236370, 797884712960

Offset: 0

Robert A. Russell, May 22 2018

Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.

			For a(6) = 3, the color patterns for both rows and loops are ABCCDE, ABCDBE, and ABCDEA.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-38,38,40,-40).

Fifth column of A304972.
Fifth column of A140735 for odd n.
Fifth column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 5 of A305008.

I:=[0,0,0,0,1,3,12]; [0] cat [n le 7 select I[n] else Self(n-1) +11*Self(n-2) -11*Self(n-3) -38*Self(n-4) +38*Self(n-5) +40*Self(n-6) -40*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

Table[If[EvenQ[n], 3 StirlingS2[n/2+2, 5] - 11 StirlingS2[n/2+1, 5] + 6 StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3 StirlingS2[(n+3)/2, 5]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 11, -11, -38, 38, 40, -40}, {0, 0, 0, 0, 1, 3, 12}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product[1 - k*x^2, {k,1,5}], {x, 0, 50}],x] (* Stefano Spezia, Oct 16 2018 *)

m=40; v=concat([0,0,0,0,1,3,12], vector(m-7)); for(n=8, m, v[n] = v[n-1] +11*v[n-2] -11*v[n-3] -38*v[n-4] +38*v[n-5] +40*v[n-6] -40*v[n-7] ); concat([0], v) \\ G. C. Greubel, Oct 16 2018

a(n) = [n==0 mod 2] * (3*S2(n/2+2, 5) - 11*S2(n/2+1, 5) + 6*S2(n/2, 5)) + [n==1 mod 2] * (S2((n+5)/2, 5) - 3*S2((n+3)/2, 5)) where S2(n,k) is the Stirling subset number A008277(n,k).
G.f.: x^5 *(1 + x)*(1 - 3*x^2)*(1 + 2*x - 2*x^2) / Product_{k=1..5} (1 - k*x^2).
a(n) = A304972(n,5).
a(2m-1) = A140735(m,5).
a(2m) = A293181(m,5).

A304976 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 18, 46, 195, 461, 1696, 3836, 13097, 28819, 94094, 203322, 644911, 1376217, 4279692, 9051592, 27755013, 58319855, 176992090, 370087718, 1114496747, 2321721493, 6950406008, 14437363668, 43021681249, 89162536011, 264732674406, 547676535634

Offset: 0

Robert A. Russell, May 22 2018

Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.

			For a(7) = 3, the color patterns for both rows and loops are ABCDCEF, ABCDEBF, and ABCDEFA.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,16,-16,-91,91,216,-216,-180, 180).

Sixth column of A304972.
Sixth column of A140735 for odd n.
Sixth column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 6 of A305008.

I:=[0,0,0,0,0,1,3,18,46]; [0] cat [n le 9 select I[n] else Self(n-1) +16*Self(n-2) -16*Self(n-3) -91*Self(n-4) +91*Self(n-5) +216*Self(n-6) -216*Self(n-7) -180*Self(n-8) +180*Self(n-9): n in [1..40]]; // G. C. Greubel, Oct 16 2018

Table[If[EvenQ[n], StirlingS2[n/2 + 3, 6] - 3 StirlingS2[n/2 + 2, 6] - 8 StirlingS2[n/2 + 1, 6] + 16 StirlingS2[n/2, 6], 3 StirlingS2[(n + 5)/2, 6] - 17 StirlingS2[(n + 3)/2, 6] + 20 StirlingS2[(n + 1)/2, 6]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 16, -16, -91, 91, 216, -216, -180, 180}, {0, 0, 0, 0, 0, 1, 3, 18, 46}, 40]] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product[1 - k*x^2, {k,1,6}], {x, 0, 50}], x] (* Stefano Spezia, Oct 20 2018 *)

m=40; v=concat([0,0,0,0,0,1,3,18,46], vector(m-9)); for(n=10, m, v[n] = v[n-1] +16*v[n-2] -16*v[n-3] -91*v[n-4] +91*v[n-5] +216*v[n-6] -216*v[n-7] -180*v[n-8] +180*v[n-9]); concat([0], v) \\ G. C. Greubel, Oct 16 2018

a(n) = [n==0 mod 2] * (S2(n/2+3, 6) - 3*S2(n/2+2, 6) - 8*S2(n/2+1, 6) + 16*S2(n/2, 6)) + [n==1 mod 2] * (3*S2((n+5)/2, 6) - 17*S2((n+3)/2, 6) + 20*S2((n+1)/2, 6 )) where S2(n,k) is the Stirling subset number A008277(n,k).
G.f.: x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2).
a(n) = A304972(n,6).
a(2m-1) = A140735(m,6).
a(2m) = A293181(m,6).

A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 0, 1, 1, 3, 10, 12, 3, 9, 3, 0, 1, 1, 7, 33, 59, 30, 67, 42, 6, 18, 4, 0, 1, 1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1, 1, 31, 333, 1187, 1365, 3112, 3675, 1596, 2700, 1655, 330, 605, 195, 15, 45, 6, 0, 1

Offset: 1

Andrew Howroyd, Oct 24 2017

T(n,k) = coefficient of x^k for A(3,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 13 2018

			Triangle begins:
  1,  0,   1;
  1,  1,   3,   2,   0,   1;
  1,  3,  10,  12,   3,   9,   3,   0,   1;
  1,  7,  33,  59,  30,  67,  42,   6,  18,   4,  0,  1;
  1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1;
  ...
Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The second row is [1, 1, 3, 2, 0, 1] because the set partitions that are invariant under this permutation in increasing order of number of parts are {{1, 2, 3, 4, 5, 6}}; {{1, 2, 3}, {4, 5, 6}}; {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}; {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1395
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums are A002874.
Column k=3 gives A053156.
Maximum row values are A294202.
Unrelated to A002875.
Cf. A002872, A002873, A293181.

T:= proc(n, k) option remember; `if`([n, k]=[0, 0], 1, 0)+
     `if`(n>0 and k>0, k*T(n-1, k)+T(n-1, k-1)+T(n-1, k-3), 0)
    end:
seq(seq(T(n, k), k=1..3*n), n=1..8);  # Alois P. Heinz, Sep 20 2019

T[n_, k_] := T[n,k] = If[n>0 && k>0, k T[n-1,k] + T[n-1,k-1] + T[n-1,k-3], Boole[n==0 && k==0]] (* modification of Gilbert & Riordan recursion *)
Table[T[n, k], {n,1,10}, {k,1,3n}] // Flatten (* Robert A. Russell, Jun 13 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k)={my(ci=PermCycleIndex(CylinderPerms(3,n)[2])); StructsByCycleIndex(ci,k) - if(k>1,StructsByCycleIndex(ci,k-1))}
for (n=1, 6, for(k=1, 3*n, print1(T(n,k), ", ")); print);

G(n)={Vec(-1+serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
{ my(A=G(6)); for(n=1, #A, print(Vecrev(A[n]/y))) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = [n==0 & k==0] + [n>0 & k>0] * (k*T(n-1,k) + T(n-1,k-1) + T(n-1,k-3)). - Robert A. Russell, Jun 13 2018

T(n,k) = n!*[x^n*y^k] exp(Sum_{d|3} y^d*(exp(d*x) - 1)/d). - Andrew Howroyd, Sep 20 2019

cp's OEIS Frontend

A304975 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 5 colors (sets).

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A304976 Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 6 colors (sets).

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A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

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