A177262 - OEIS

A177262

Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} starting with exactly k consecutive integers (1<=k<=n).

1, 1, 1, 4, 1, 1, 18, 4, 1, 1, 96, 18, 4, 1, 1, 600, 96, 18, 4, 1, 1, 4320, 600, 96, 18, 4, 1, 1, 35280, 4320, 600, 96, 18, 4, 1, 1, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1, 36288000, 3265920, 322560, 35280, 4320, 600, 96, 18, 4, 1, 1

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OFFSET

1,4

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

FORMULA

T(n, k) = (n-k)*(n-k)! if k < n, otherwise T(n,n) = 1.

T(n, 1) = A094258(n) = (n-1)!(n-1).

Sum_{k=1..n} T(n, k) = A000142(n) (row sums).

Sum_{k=1..n} k*T(n,k) = Sum_{j=1..n} j! = A007489(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A005165(n). - G. C. Greubel, May 18 2024

EXAMPLE

T(4,2)=4 because we have 1243, 2314, 3412, and 3421.

Triangle starts:

1, 1;

4, 1, 1;

18, 4, 1, 1;

96, 18, 4, 1, 1;

600, 96, 18, 4, 1, 1;

4320, 600, 96, 18, 4, 1, 1;

35280, 4320, 600, 96, 18, 4, 1, 1;

MAPLE

T := proc (n, k) if k = n then 1 elif k < n then factorial(n-k)*(n-k) else 0 end if end proc: for n to 11 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

MATHEMATICA

T[n_, k_]:= (n-k+1)! -(n-k)! +Boole[k==n];

Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, May 18 2024 *)

PROG

(Magma)

A177262:= func< n, k | k eq n select 1 else (n-k)*Factorial(n-k) >;

[A177262(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, May 18 2024

(SageMath)

def A177262(n, k): return (n-k)*factorial(n-k) + int(k==n)

flatten([[A177262(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, May 18 2024

CROSSREFS

Cf. A000142 (row sums), A005165, A007489, A094258.