A395326 - OEIS

%I #12 Apr 27 2026 10:25:41

%S 1,0,2,0,3,4,0,5,18,8,0,9,67,72,16,0,17,240,470,240,32,0,33,859,2745,

%T 2420,720,64,0,65,3108,15323,20790,10360,2016,128,0,129,11407,84042,

%U 165081,122640,39200,5376,256,0,257,42480,458900,1257480,1307922,610848,135744,13824,512

%N Triangle read by rows: T(n,k) is the number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles with k orbits of blocks.

%H Alois P. Heinz, <a href="/A395326/b395326.txt">Rows n = 0..150, flattened</a>

%F E.g.f.: exp( y*((exp(2*x) - 1)/2 + exp(x) - 1 ) ).

%e Triangle begins:

%e   1;

%e   0,   2;

%e   0,   3,     4;

%e   0,   5,    18,     8;

%e   0,   9,    67,    72,     16;

%e   0,  17,   240,   470,    240,     32;

%e   0,  33,   859,  2745,   2420,    720,    64;

%e   0,  65,  3108, 15323,  20790,  10360,  2016,  128;

%e   0, 129, 11407, 84042, 165081, 122640, 39200, 5376, 256;

%e   ...

%p b:= proc(n) option remember; `if`(n=0, 1, add((1+

%p       2^(j-1))*binomial(n-1, j-1)*expand(b(n-j)*x), j=1..n))

%p     end:

%p T:= (n,k)-> coeff(b(n), x, k):

%p seq(seq(T(n,k), k=0..n), n=0..9);  # _Alois P. Heinz_, Apr 27 2026

%o (PARI) T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp( y*((exp(2*x + O(x*x^n)) - 1)/2 + exp(x + O(x*x^n)) - 1 ) )))]}

%o { my(A=T(8)); for(i=1, #A, print(A[i])) }

%Y Row sums are A002872.

%Y Main diagonal is A000079.

%Y Column k=1 is A000051(n-1).

%Y Cf. A293181, A395327.

%K nonn,tabl

%O 0,3

%A _Andrew Howroyd_ and _Andrei Zabolotskii_, Apr 19 2026