A375854 - OEIS

%I #11 Nov 13 2024 20:34:18

%S 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 380,2248,12032,58576,1,9,74,554,3768,23272,130768,671568,1,10,92,776,

%U 5984,42112,270400,1586944,8546432,1,11,112,1052,9088,72032,523072,3479744,21241984,119401856

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

%F T(n, k) = Sum_{j=0..k} 2^(k - j)*binomial(n, j)*binomial(k, j)*j!.

%e Triangle starts:

%e [0] 1;

%e [1] 1, 3;

%e [2] 1, 4, 14;

%e [3] 1, 5, 22, 86;

%e [4] 1, 6, 32, 152, 648;

%e [5] 1, 7, 44, 248, 1256, 5752;

%e [6] 1, 8, 58, 380, 2248, 12032, 58576;

%e [7] 1, 9, 74, 554, 3768, 23272, 130768, 671568;

%e [8] 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432;

%e [9] 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856;

%e ...

%p T := (n, k) -> 2^k * hypergeom([-n, -k], [], 1/2):

%p for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # _Peter Luschny_, Sep 02 2024

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

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

%o (Python)

%o from math import isqrt, comb, factorial

%o def A375854(n):

%o     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))

%o     b = n-comb(a+1,2)

%o     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 Cf. A375855, A000012, A087912 (main diagonal).

%K nonn,tabl

%O 0,3

%A _Detlef Meya_, Aug 31 2024