A110142 -id:A110142 - cp's OEIS frontend

A110144 Terms of A110142 at positions p(n)+1, where p(n) = A000041(n) is the number of partitions of n; a(n) = A110142(p(n)+1) for n>=1, with a(0) = 1.

1, 2, 3, 8, 6, 48, 24, 384, 144, 3840, 1152, 46080, 11520, 645120, 138240, 10321920, 1935360, 185794560, 30965760, 3715891200, 557383680, 81749606400, 11147673600, 1961990553600, 245248819200, 51011754393600, 5885971660800

Offset: 0

Paul D. Hanna, Jul 14 2005

nonn

Cf. A110142, A110141, A000041, A000165 (double factorials).

a(n)=if(n==0,1,if(n%2==1,2^(n\2+1)*(n\2+1)!,3*2^((n-1)\2)*((n-1)\2)!))

a(2*n+1) = 2^n*n!, a(2*n+2) = 3*2^n*n! for n>0, with a(0) = 1.

A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

Original entry on oeis.org

1, 1, 2, 2, 6, 2, 3, 24, 4, 3, 8, 4, 120, 12, 6, 8, 4, 6, 5, 720, 48, 18, 16, 8, 6, 5, 48, 8, 18, 6, 5040, 240, 72, 48, 24, 12, 10, 48, 8, 18, 6, 24, 10, 12, 7, 40320, 1440, 360, 192, 96, 36, 30, 96, 16, 36, 12, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 362880, 10080, 2160, 960

Offset: 0

Paul D. Hanna, Jul 13 2005

Row n starts with n!, after which the following pattern holds. When terms of row n are divided by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1, the result is the initial terms of A110142. E.g., row 6 is: {720,48,18,16,8,6,5,48,8,18,6}; divide by respective factorials: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!} with {4!,3!,2!,1!,0!} respectively occurring {1,1,2,2,4} times (A002865), yields the initial terms of A110142: {1,2,3,8,4,6,5,48,8,18,6}.
The term of the sequence corresponding to the product c_1^{n_1}c_2^{n_2}...c_k^{n_k} is equal to the number of elements in the centralizer of a permutation of n_1+2n_2+...+kn_k elements whose cycle type is 1^{n_1}2^{n_2}...k^{n^k}. (This fact is very standard, in particular, for the theory of symmetric functions.) - Vladimir Dotsenko, Apr 19 2009
Multiplying the values of row n by the corresponding values in row n of A102189, one obtains n!. - Jaimal Ichharam, Aug 06 2015
a(n,k) is the number of permutations in S_n that commute with a permutation having cycle type "k".  Here, the cycle type of an n-permutation pi is the vector (i_1,...,i_n) where i_j is the number of cycles in pi of length j.  These A000041(n) vectors can be ordered in reverse lexicographic order.  The k-th cycle type is the k-th vector in this ordering. - Geoffrey Critzer, Jan 18 2019

			Coefficients [x^n] exp(c1*x + (c2/2)*x^2 + (c3/3)*x^3 + ...) begin:
[x^0]: 1;
[x^1]: 1*c1;
[x^2]: (1/2)*c1^2 + (1/2)*c2;
[x^3]: (1/6)*c1^3 + (1/2)*c1*c2 + (1/3)*c3;
[x^4]: (1/24)*c1^4 + (1/4)*c1^2*c2 + (1/3)*c1*c3 + (1/8)*c2^2 + (1/4)*c4;
[x^5]: (1/120)*c1^5 + (1/12)*c1^3*c2 + (1/6)*c1^2*c3 + (1/8)*c1*c2^2 + (1/4)*c1*c4 + (1/6)*c2*c3 + (1/5)*c5;
[x^6]: (1/720)*c1^6 + (1/48)*c1^4*c2 + (1/18)*c1^3*c3 + (1/16)*c1^2*c2^2 + (1/8)*c1^2*c4 + (1/6)*c1*c2*c3 + (1/5)*c1*c5 + (1/48)*c2^3 + (1/8)*c2*c4 + (1/18)*c3^2 + (1/6)*c6;
forming this triangle of unit fraction coefficients:
1;
1;
2,2;
6,2,3;
24,4,3,8,4;
120,12,6,8,4,6,5;
720,48,18,16,8,6,5,48,8,18,6;
5040,240,72,48,24,12,10,48,8,18,6,24,10,12,7;
40320,1440,360,192,96,36,30,96,16,36,12,24,10,12,7,384,32,36,12,15,32,8;
362880,10080,2160,960,480,144,120,288,48,108,36,48,20,24,14,384,32,36,12,15,32,8,144,40,24,14,162,18,20,9; ...

Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford University Press, 1995. [From Vladimir Dotsenko, Apr 19 2009]

Cf. A000041, A002865, A102189, A110142, A110143 (row sums).

First, second and third entries of each row are given (up to an offset) by A000142, A052849, and A052560 respectively. - Vladimir Dotsenko, Apr 19 2009

Table[n!/CoefficientRules[n! CycleIndex[SymmetricGroup[n], s]][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 18 2019 *)

Number of terms in row n is A000041(n) (partition numbers). The unit fractions of each row sum to unity: Sum_{k=1..A000041(n)} 1/T(n, k) = 1.

a(n,k) = n!/A181897(n,k). - Geoffrey Critzer, Jan 18 2019

cp's OEIS Frontend

A110144 Terms of A110142 at positions p(n)+1, where p(n) = A000041(n) is the number of partitions of n; a(n) = A110142(p(n)+1) for n>=1, with a(0) = 1.

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A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

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