A100326 Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).

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

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

Cf. A003169, A100324, A100327 (row sums), A158687, A264717 (central terms).

Sequence in context: A114189 A200659 A059110 * A303728 A321627 A350557

Adjacent sequences: A100323 A100324 A100325 * A100327 A100328 A100329

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 17 2004

Status

approved