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A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).

%I A123515 #9 Jan 16 2022 11:08:59
%S A123515 1,0,2,2,0,3,0,8,0,4,5,0,18,0,5,0,26,0,32,0,6,12,0,75,0,50,0,7,0,76,0,
%T A123515 164,0,72,0,8,28,0,264,0,305,0,98,0,9,0,208,0,680,0,510,0,128,0,10,64,
%U A123515 0,840,0,1460,0,791,0,162,0,11,0,544,0,2480,0,2772,0,1160,0,200,0,12
%N A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).
%C A123515 Also the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 312 exactly once (n>=4, 2<=k<=n-2). Example: T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 312: 523, 524 and 534).
%H A123515 G. C. Greubel, <a href="/A123515/b123515.txt">Rows n = 4..54 of the triangle, flattened</a>
%H A123515 E. Deutsch, A. Robertson and D. Saracino, <a href="https://doi.org/10.1016/j.ejc.2004.09.006">Refined restricted involutions</a>, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 492 and 498).
%F A123515 T(n, k) = 2^((n-k-6)/2)*(k-1)*( binomial((n+k)/2-2, (n-k)/2-1) + 2*binomial((n+k)/2-3, (n-k)/2-1) + binomial((n+k)/2-4, (n-k)/2-1) ) for n>=4, n+k even; T(n,k) = 0 otherwise.
%F A123515 From _G. C. Greubel_, Jan 16 2022: (Start)
%F A123515 Sum_{k=2..n-4} T(n, k) = A045623(n).
%F A123515 Sum_{k=2..floor(n/2)} T(n-k+2, k) = (1/9)*[n=4] + (1+(-1)^n)*n*3^((n-8)/2). (End)
%e A123515 T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 231: 231, 241 and 341).
%e A123515 Triangle starts:
%e A123515    1;
%e A123515    0,   2;
%e A123515    2,   0,   3;
%e A123515    0,   8,   0,    4;
%e A123515    5,   0,  18,    0,    5;
%e A123515    0,  26,   0,   32,    0,    6;
%e A123515   12,   0,  75,    0,   50,    0,   7;
%e A123515    0,  76,   0,  164,    0,   72,   0,    8;
%e A123515   28,   0, 264,    0,  305,    0,  98,    0,   9;
%e A123515    0, 208,   0,  680,    0,  510,   0,  128,   0,  10;
%e A123515   64,   0, 840,    0, 1460,    0, 791,    0, 162,   0, 11;
%e A123515    0, 544,   0, 2480,    0, 2772,   0, 1160,   0, 200,  0, 12;
%p A123515 T:=proc(n,k) if n>=4 and n+k mod 2 = 0 then (k-1)*2^((n-k-6)/2)*(binomial((n+k)/2-2,(n-k)/2-1)+2*binomial((n+k)/2-3, (n-k)/2-1)+binomial((n+k)/2-4,(n-k)/2-1)) else 0 fi end: for n from 4 to 16 do seq(T(n,k),k=2..n-2) od; # yields sequence in triangular form
%t A123515 T[n_, k_]:= ((1+(-1)^(n-k))/2)*2^((n-k-6)/2)*(k-1)* Sum[Binomial[2, j]*
%t A123515   Binomial[(n+k-2*(j+2))/2, (n-k-2)/2], {j, 0, 2}];
%t A123515 Table[T[n, k], {n,4,16}, {k,2,n-2}]//Flatten (* _G. C. Greubel_, Jan 16 2022 *)
%o A123515 (Sage)
%o A123515 def A123515(n,k): return ((1+(-1)^(n+k))/2)*2^((n-k-6)/2)*(k-1)*sum( binomial(2, j)*binomial((n+k-2*j-2)/2, (n-k-2)/2) for j in (0..2) )
%o A123515 flatten([[A123515(n,k) for k in (2..n-2)] for n in (4..16)]) # _G. C. Greubel_, Jan 16 2022
%Y A123515 Cf. A045623, A120926, A123514, A112554.
%K A123515 nonn,tabl
%O A123515 4,3
%A A123515 _Emeric Deutsch_, Oct 13 2006