A099759 Triangle read by rows: T(n,0)=1, T(n,n)=1, T(n, k) = 2*(n-k)*T(n-1, k-1) + 2*k*T(n-1, k).

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 30, 96, 30, 1, 1, 68, 564, 564, 68, 1, 1, 146, 2800, 6768, 2800, 146, 1, 1, 304, 12660, 63008, 63008, 12660, 304, 1, 1, 622, 54288, 504648, 1008128, 504648, 54288, 622, 1, 1, 1260, 225860, 3679344, 13111504, 13111504, 3679344, 225860, 1260, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

Sum_{k=0..n} T(n, k) = 2*(n-1)*(Sum_{k=0..n-1} T(n-1, k)) + 2 = A099760(n).

Example

Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1, 12,  12,   1;
  1, 30,  96,  30,  1;
  1, 68, 564, 564, 68, 1;

Maple

T:=proc(n, k) if k=0 or n=k then 1 elif k>n then 0 else 2*(n-k)*T(n-1, k-1)+2*k*T(n-1, k) fi end: for n from 0 to 9 do [seq(T(n, k), k=0..n)] od; # gives the triangle row by row # Emeric Deutsch, Nov 16 2004

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, 2*(n-k)*T[n-1, k-1] +2*k*T[n-1, k]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 03 2019 *)

Prog

(PARI) T(n, k) = if(k==0 || k==n, 1, 2*(n-k)*T(n-1, k-1) + 2*k*T(n-1, k));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 03 2019
(Sage)
def T(n, k):
    if (k==0 or k==n): return 1
    else: return 2*k*T(n-1, k) + 2*(n-k)* T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..9)] # G. C. Greubel, Sep 03 2019
(GAP)
T:= function(n, k)
    if k=0 or k=n then return 1;
    else return 2*(n-k)*T(n-1, k-1) + 2*k*T(n-1, k);
    fi;
  end;
Flat(List([0..9], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Sep 03 2019

Crossrefs

Cf. A060187, A099760.

Sequence in context: A080416 A213166 A168619 * A350819 A072590 A350745

Adjacent sequences: A099756 A099757 A099758 * A099760 A099761 A099762

Keyword

easy,tabl,nonn

Author

Miklos Kristof, Nov 11 2004

Extensions

More terms from Emeric Deutsch, Nov 16 2004

Status

approved