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A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).

%I A375854 #11 Nov 13 2024 20:34:18
%S A375854 1,1,3,1,4,14,1,5,22,86,1,6,32,152,648,1,7,44,248,1256,5752,1,8,58,
%T A375854 380,2248,12032,58576,1,9,74,554,3768,23272,130768,671568,1,10,92,776,
%U A375854 5984,42112,270400,1586944,8546432,1,11,112,1052,9088,72032,523072,3479744,21241984,119401856
%N A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).
%F A375854 T(n, k) = Sum_{j=0..k} 2^(k - j)*binomial(n, j)*binomial(k, j)*j!.
%e A375854 Triangle starts:
%e A375854 [0] 1;
%e A375854 [1] 1, 3;
%e A375854 [2] 1, 4, 14;
%e A375854 [3] 1, 5, 22, 86;
%e A375854 [4] 1, 6, 32, 152, 648;
%e A375854 [5] 1, 7, 44, 248, 1256, 5752;
%e A375854 [6] 1, 8, 58, 380, 2248, 12032, 58576;
%e A375854 [7] 1, 9, 74, 554, 3768, 23272, 130768, 671568;
%e A375854 [8] 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432;
%e A375854 [9] 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856;
%e A375854 ...
%p A375854 T := (n, k) -> 2^k * hypergeom([-n, -k], [], 1/2):
%p A375854 for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # _Peter Luschny_, Sep 02 2024
%t A375854 T[n_, k_] := Sum[2^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
%t A375854 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%o A375854 (Python)
%o A375854 from math import isqrt, comb, factorial
%o A375854 def A375854(n):
%o A375854     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A375854     b = n-comb(a+1,2)
%o A375854     return sum(comb(a,j)*comb(b,j)*factorial(j)<<b-j for j in range(b+1)) # _Chai Wah Wu_, Nov 13 2024
%Y A375854 Cf. A375855, A000012, A087912 (main diagonal).
%K A375854 nonn,tabl
%O A375854 0,3
%A A375854 _Detlef Meya_, Aug 31 2024