A374427 Triangle read by rows: T(n, k) = n! * 2^k * hypergeom([-k], [-n], -1/2).

1, 1, 1, 2, 3, 5, 6, 10, 17, 29, 24, 42, 74, 131, 233, 120, 216, 390, 706, 1281, 2329, 720, 1320, 2424, 4458, 8210, 15139, 27949, 5040, 9360, 17400, 32376, 60294, 112378, 209617, 391285, 40320, 75600, 141840, 266280, 500184, 940074, 1767770, 3325923, 6260561

Offset: 0

Author

Peter Luschny, Jul 28 2024

Keywords

• nonn
• tabl

Examples

			     1
     1      1
     2      3      5
     6     10     17     29
    24     42     74    131    233
   120    216    390    706   1281   2329
   720   1320   2424   4458   8210  15139  27949
  5040   9360  17400  32376  60294 112378 209617 391285
 40320  75600 141840 266280 500184 940074 1767770 3325923 6260561
362880 685440 1295280 2448720 4631160 8762136 16584198 31400626 59475329
		

Crossrefs

Cf. A000354 (main diagonal), A374428, A007680 (col k=0).

Programs

Maple

A374427 := proc(n,k)
    (-1)^k*add((-2)^(k-j)*binomial(k,k-j)*(n-j)!,j=0..k) ;
end proc:
seq(seq(A374427(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 30 2024

Mathematica

T[n_, k_] := n! 2^k Hypergeometric1F1[-k, -n, -1/2];
(* Alternative: )
T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Detlef Meya, Aug 12 2024 *)

Formula

T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(k, k - j)*(n - j)!. - Detlef Meya, Aug 12 2024