OFFSET
0,5
COMMENTS
Row n contains 1 + floor(n/2) entries.
Sum of entries in row n = n! (A000142).
LINKS
Seiichi Manyama, Rows n = 0..200, flattened
R. A. Brualdi and E. Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.
FORMULA
T(n, k) = Sum_{j=0..floor(n/2)} (-1)^(k+j)*binomial(j,k)*(n-j)!/j!.
T(n, 0) = A177249(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = (n-1)! (n >= 2).
G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+k) / (1+x^2)^(j+1). - Seiichi Manyama, Feb 24 2024
EXAMPLE
T(5,2)=3 because we have (12)(34)(5), (12)(3)(45), and (1)(23)(45).
Triangle starts:
1;
1;
1, 1;
4, 2;
19, 4, 1;
99, 18, 3;
611, 99, 9, 1;
MAPLE
T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-j)/factorial(j), j = 0 .. floor((1/2)*n)) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
MATHEMATICA
T[n_, k_] := Sum[(-1)^(k + j)*Binomial[j, k]*(n - j)!/j!, {j, 0, n/2}];
Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)
PROG
(PARI) T(n, k) = sum(j=0, n\2, (-1)^(k+j)*binomial(j, k)*(n-j)!/j!);
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Nov 21 2017
(Magma)
F:=Factorial;
A177248:= func< n, k | (&+[(-1)^j*F(n-k-j)/(F(k)*F(j)): j in [0..Floor((n-2*k)/2)]]) >;
[A177248(n, k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Apr 28 2024
(SageMath)
f=factorial;
def A177248(n, k): return sum((-1)^j*f(n-k-j)/(f(k)*f(j)) for j in range(1+(n-2*k)//2))
flatten([[A177248(n, k) for k in range(1+n//2)] for n in range(17)]) # G. C. Greubel, Apr 28 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 07 2010
STATUS
approved