OFFSET
0,4
COMMENTS
The leftmost column equals A003169 shift one place right.
Each column k > 0 equals the convolution of the prior column and A003169.
Row sums form A100327.
The elements of the matrix inverse are T^(-1)(n,k) = (-1)^(n+k) * A158687(n,k). - R. J. Mathar, Mar 15 2013
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
FORMULA
T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1.
T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n > 0.
T(2*n, n) = A264717(n).
Sum_{k=0..n} T(n, k) = A100327(n).
G.f.: A(x, y) = (1 + G(x))/(1 - y*G(x)), where G(x) is the g.f. of A003169.
From G. C. Greubel, Jan 30 2023: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A033999(n). (End)
EXAMPLE
Rows begin:
1;
1, 1;
3, 4, 1;
14, 20, 7, 1;
79, 116, 46, 10, 1;
494, 736, 311, 81, 13, 1;
3294, 4952, 2174, 626, 125, 16, 1;
22952, 34716, 15634, 4798, 1088, 178, 19, 1;
165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1;
1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25, 1;
...
First column forms A003169 shift right.
Binomial transform of row 3 forms column 3 of square A100324: BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].
Binomial transform of row 4 forms column 4 of square A100324: BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].
MAPLE
A100326 := proc(n, k)
if k < 0 or k > n then
0 ;
elif n = 0 then
1 ;
elif k = 0 then
A003169(n)
else
add(procname(i+1, 0)*procname(n-i-1, k-1), i=0..n-k) ;
end if;
end proc: # R. J. Mathar, Mar 15 2013
MATHEMATICA
lim= 9; t[0, 0]=1; t[n_, 0]:= t[n, 0]= Sum[(k+1)*t[n-1, k], {k, 0, n-1}]; t[n_, k_]:= t[n, k]= Sum[t[j+1, 0]*t[n-j-1, k-1], {j, 0, n-k}]; Flatten[Table[t[n, k], {n, 0, lim}, {k, 0, n}]] (* Jean-François Alcover, Sep 20 2011 *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==0, 1, if(k==0, sum(i=0, n-1, (i+1)*T(n-1, i)), sum(i=0, n-k, T(i+1, 0)*T(n-i-1, k-1))); ))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(Haskell)
import Data.List (transpose)
a100326 n k = a100326_tabl !! n !! k
a100326_row n = a100326_tabl !! n
a100326_tabl = [1] : f [[1]] where
f xss@(xs:_) = ys : f (ys : xss) where
ys = y : map (sum . zipWith (*) (zs ++ [y])) (map reverse zss)
y = sum $ zipWith (*) [1..] xs
zss@((_:zs):_) = transpose $ reverse xss
-- Reinhard Zumkeller, Nov 21 2015
(SageMath)
@CachedFunction
def T(n, k): # T = A100326
if (k<0 or k>n): return 0
elif (k==n): return 1
elif (k==0): return sum((j+1)*T(n-1, j) for j in range(n))
else: return sum(T(j+1, 0)*T(n-j-1, k-1) for j in range(n-k+1))
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 30 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Nov 17 2004
STATUS
approved