A372260 Triangle read by rows: T(n, k) = (T(n-1, k-1) + T(n-1, k)) * 2 * n with initial values T(n, 0) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A001147(i) and T(i, j) = 0 if j > i.

1, 0, 2, 2, 8, 8, 8, 60, 96, 48, 60, 544, 1248, 1152, 384, 544, 6040, 17920, 24000, 15360, 3840, 6040, 79008, 287520, 503040, 472320, 230400, 46080, 79008, 1190672, 5131392, 11067840, 13655040, 9838080, 3870720, 645120, 1190672, 20314880, 101153024, 259187712, 395566080, 375889920, 219340800, 72253440, 10321920

Offset: 0

Author

Werner Schulte, Apr 24 2024

Keywords

• nonn
• easy
• tabl

Examples

			Triangle T(n, k) starts:
n\k :     0       1       2        3        4       5       6      7
====================================================================
  0 :     1
  1 :     0       2
  2 :     2       8       8
  3 :     8      60      96       48
  4 :    60     544    1248     1152      384
  5 :   544    6040   17920    24000    15360    3840
  6 :  6040   79008  287520   503040   472320  230400   46080
  7 : 79008 1190672 5131392 11067840 13655040 9838080 3870720 645120
  etc.
		

Crossrefs

Cf. A053871 (column 0), 2*A179540 (column 1), A000165 (main diagonal).

Cf. A001147.

Programs

Maple

T := proc(n, k) option remember; `if`(k > n, 0, `if`(k = n, 2^n * n!, `if`(k = 0, `if`(n < 2, 1 - n, (2*n - 2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Apr 25 2024

Mathematica

T[n_,k_]:=n!SeriesCoefficient[(Exp[-t]/Sqrt[1 - 2*t])*(2*t/(1-2*t))^k,{t,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Apr 25 2024 *)

PARI

{  T(n, k) = if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1-n,
         (2*n-2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n)))  }

PARI

memo = Map(); memoize(f, A[..]) =
{ my(res);
  if(!mapisdefined(memo, [f, A], &res), res = call(f, A);
  mapput(memo, [f, A], res)); res; }
T(n, k) =
{ if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1 - n,
  (2 * n - 2) * (memoize(T, n-1, k) + memoize(T, n-2, k))),
  (memoize(T, n-1, k-1) + memoize(T, n-1, k)) * 2 * n))); }

Formula

T(n, 0) = (2*n - 2) * (T(n-1, 0) + T(n-2, 0)) for n > 1 with initial values T(n, 0) = 1 - n for n < 2 (see A053871).

T(n, k) = (Sum_{i=0..k} binomial(k, i) * T(n-i, 0)) * 2^(2*k) * binomial(n, k) / binomial(2*k, k).

E.g.f. of column k: (exp(-t) / sqrt(1 - 2*t)) * (2*t / (1 - 2*t))^k.

E.g.f.: exp((2*x / (1 - 2*t) - 1) * t) / sqrt(1 - 2*t).