A374427 - OEIS

%I #15 Aug 30 2024 10:41:31

%S 1,1,1,2,3,5,6,10,17,29,24,42,74,131,233,120,216,390,706,1281,2329,

%T 720,1320,2424,4458,8210,15139,27949,5040,9360,17400,32376,60294,

%U 112378,209617,391285,40320,75600,141840,266280,500184,940074,1767770,3325923,6260561

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

%F T(n, k) = (-1)^k*Sum_{j=0..k} (-2)^(k - j)*binomial(k, k - j)*(n - j)!. - _Detlef Meya_, Aug 12 2024

%e      1

%e      1      1

%e      2      3      5

%e      6     10     17     29

%e     24     42     74    131    233

%e    120    216    390    706   1281   2329

%e    720   1320   2424   4458   8210  15139  27949

%e   5040   9360  17400  32376  60294 112378 209617 391285

%e  40320  75600 141840 266280 500184 940074 1767770 3325923 6260561

%e 362880 685440 1295280 2448720 4631160 8762136 16584198 31400626 59475329

%p A374427 := proc(n,k)

%p     (-1)^k*add((-2)^(k-j)*binomial(k,k-j)*(n-j)!,j=0..k) ;

%p end proc:

%p seq(seq(A374427(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Aug 30 2024

%t T[n_, k_] := n! 2^k Hypergeometric1F1[-k, -n, -1/2];

%t (* Alternative: )

%t T[n_, k_] := (-1)^k*Sum[(-2)^(k - j)*Binomial[k, k - j]*((n - j)!), {j, 0, k}];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Detlef Meya_, Aug 12 2024 *)

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Jul 28 2024