A099755 Triangle read by rows: T(n,0)=1, T(n,n)=(2*n-1)!!+1, T(n,k) = 2*(n-k) * T(n-1,k-1) + 2*(k+1)*T(n-1,k).

1, 1, 2, 1, 10, 4, 1, 44, 44, 16, 1, 182, 440, 216, 106, 1, 736, 3732, 3488, 1492, 946, 1, 2954, 28280, 50296, 28872, 14336, 10396, 1, 11828, 199220, 628608, 590496, 287520, 174216, 135136, 1, 47326, 1337256, 7021064, 10933824, 6993216, 3589104, 2510608, 2027026

Offset

0,3

Links

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

Formula

Sum_{k=0..n} T(n, k) = (2*k+1)!! = (2*k+1)*(2*k-1)*(2*k-3)*...

Example

Triangle begins:
  1;
  1,   2;
  1,  10    4;
  1,  44,  44,  16;
  1, 182, 440, 216, 106;

Maple

T:=proc(n, k) if k=0 then 1 elif n=k then 1+(2*k)!/(k!*2^k) elif k>n then 0 else 2*(n-k)*T(n-1, k-1)+(2*k+2)*T(n-1, k) fi end: for n from 0 to 9 do [seq(T(n, k), k=0..n)] od;

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n, (2*n-1)!! +1, 2*(n-k)*T[n-1, k-1] + 2*(k+1)*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, 1, if(k==n, (2*n)!/(2^n*n!) + 1, 2*(n-k)*T(n-1, k-1) + 2*(k+1)*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): return 1
    elif (k==n): return factorial(2*n)/(2^n*factorial(n)) + 1
    else: return 2*(k+1)*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 then return 1;
    elif k=n then return Factorial(2*n)/(2^n*Factorial(n)) + 1;
    else return 2*(n-k)*T(n-1, k-1) + 2*(k+1)*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.

Sequence in context: A225911 A163235 A142963 * A202483 A110682 A110327

Adjacent sequences: A099752 A099753 A099754 * A099756 A099757 A099758

Keyword

easy,nonn,tabl

Author

Miklos Kristof, Nov 11 2004

Extensions

Name corrected by G. C. Greubel, Sep 04 2019

Status

approved