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A123514 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 321 exactly once (n>=3; 1<=k<=n-2).

%I A123514 #11 Jan 21 2022 07:54:03
%S A123514 1,0,2,4,0,3,0,10,0,4,14,0,18,0,5,0,40,0,28,0,6,48,0,81,0,40,0,7,0,
%T A123514 150,0,140,0,54,0,8,165,0,330,0,220,0,70,0,9,0,550,0,616,0,324,0,88,0,
%U A123514 10,572,0,1287,0,1040,0,455,0,108,0,11,0,2002,0,2548,0,1638,0,616,0,130,0,12
%N A123514 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 321 exactly once (n>=3; 1<=k<=n-2).
%H A123514 G. C. Greubel, <a href="/A123514/b123514.txt">Rows n = 3..53 of the triangle, flattened</a>
%H A123514 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. 493 and 498).
%F A123514 T(n,k) = k*(k+3)*binomial(n+1,(n-k-2)/2)/(n+1), for n>=3, 1<=k<=n-2, n-k even.
%F A123514 From _G. C. Greubel_, Jan 15 2022: (Start)
%F A123514 Sum_{k=1..n-2} T(n, k) = A191389(n+1).
%F A123514 Sum_{k=1..floor((n-1)/2)} T(n-k, k) = ((1-(-1)^n)/2)*(12/(n+9))*binomial(n+2, (n- 3)/2). (End)
%e A123514 T(4,2)=2 because we have 1432 and 3214 (also 4231 is an involution with 2 fixed points but contains twice the pattern 321: 421 and 431).
%e A123514 Triangle starts:
%e A123514     1;
%e A123514     0,   2;
%e A123514     4,   0,    3;
%e A123514     0,  10,    0,   4;
%e A123514    14,   0,   18,   0,    5;
%e A123514     0,  40,    0,  28,    0,   6;
%e A123514    48,   0,   81,   0,   40,   0,   7;
%e A123514     0, 150,    0, 140,    0,  54,   0,  8;
%e A123514   165,   0,  330,   0,  220,   0,  70,  0,   9;
%e A123514     0, 550,    0, 616,    0, 324,   0, 88,   0, 10;
%e A123514   572,   0, 1287,   0, 1040,   0, 455,  0, 108,  0, 11;
%p A123514 T:=proc(n,k) if n-k mod 2 = 0 and k<=n then k*(k+3)*binomial(n+1,(n-k)/2-1)/(n+1) else 0 fi end: for n from 3 to 15 do seq(T(n,k),k=1..n-2) od; # yields sequence in triangular form
%t A123514 T[n_, k_]:= ((1+(-1)^(n-k))/2)*k*(k+3)*Binomial[n+1, (n-k-2)/2]/(n+1);
%t A123514 Table[T[n, k], {n, 3, 15}, {k, n-2}]//Flatten (* _G. C. Greubel_, Jan 15 2022 *)
%o A123514 (Magma)
%o A123514 A123514:= func< n,k | ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*Binomial(n+1, Floor((n-k-2)/2)) >;
%o A123514 [A123514(n,k): k in [1..n-2], n in [3..15]]; // _G. C. Greubel_, Jan 15 2022
%o A123514 (Sage)
%o A123514 def A123514(n,k): return ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*binomial(n+1, (n-k-2)//2)
%o A123514 flatten([[A123514(n,k) for k in (1..n-2)] for n in (3..15)]) # _G. C. Greubel_, Jan 15 2022
%Y A123514 Cf. A003517, A112554, A123515, A191389.
%K A123514 nonn,tabl
%O A123514 3,3
%A A123514 _Emeric Deutsch_, Oct 13 2006