OFFSET
3,3
LINKS
G. C. Greubel, Rows n = 3..53 of the triangle, flattened
E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 493 and 498).
FORMULA
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.
From G. C. Greubel, Jan 15 2022: (Start)
Sum_{k=1..n-2} T(n, k) = A191389(n+1).
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)
EXAMPLE
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).
Triangle starts:
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, 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, 10;
572, 0, 1287, 0, 1040, 0, 455, 0, 108, 0, 11;
MAPLE
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
MATHEMATICA
T[n_, k_]:= ((1+(-1)^(n-k))/2)*k*(k+3)*Binomial[n+1, (n-k-2)/2]/(n+1);
Table[T[n, k], {n, 3, 15}, {k, n-2}]//Flatten (* G. C. Greubel, Jan 15 2022 *)
PROG
(Magma)
A123514:= func< n, k | ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*Binomial(n+1, Floor((n-k-2)/2)) >;
[A123514(n, k): k in [1..n-2], n in [3..15]]; // G. C. Greubel, Jan 15 2022
(Sage)
def A123514(n, k): return ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*binomial(n+1, (n-k-2)//2)
flatten([[A123514(n, k) for k in (1..n-2)] for n in (3..15)]) # G. C. Greubel, Jan 15 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Oct 13 2006
STATUS
approved