%I #33 Jan 31 2023 08:47:50
%S 1,1,1,3,4,1,14,20,7,1,79,116,46,10,1,494,736,311,81,13,1,3294,4952,
%T 2174,626,125,16,1,22952,34716,15634,4798,1088,178,19,1,165127,250868,
%U 115048,36896,9094,1724,240,22,1,1217270,1855520,862607,285689,74687,15629,2561,311,25,1
%N 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).
%C The leftmost column equals A003169 shift one place right.
%C Each column k > 0 equals the convolution of the prior column and A003169.
%C Row sums form A100327.
%C The elements of the matrix inverse are T^(-1)(n,k) = (-1)^(n+k) * A158687(n,k). - _R. J. Mathar_, Mar 15 2013
%H Reinhard Zumkeller, <a href="/A100326/b100326.txt">Rows n = 0..120 of triangle, flattened</a>
%F T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1.
%F T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n > 0.
%F T(2*n, n) = A264717(n).
%F Sum_{k=0..n} T(n, k) = A100327(n).
%F G.f.: A(x, y) = (1 + G(x))/(1 - y*G(x)), where G(x) is the g.f. of A003169.
%F From _G. C. Greubel_, Jan 30 2023: (Start)
%F Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
%F Sum_{k=0..n-1} (-1)^k*T(n, k) = A033999(n). (End)
%e Rows begin:
%e 1;
%e 1, 1;
%e 3, 4, 1;
%e 14, 20, 7, 1;
%e 79, 116, 46, 10, 1;
%e 494, 736, 311, 81, 13, 1;
%e 3294, 4952, 2174, 626, 125, 16, 1;
%e 22952, 34716, 15634, 4798, 1088, 178, 19, 1;
%e 165127, 250868, 115048, 36896, 9094, 1724, 240, 22, 1;
%e 1217270, 1855520, 862607, 285689, 74687, 15629, 2561, 311, 25, 1;
%e ...
%e First column forms A003169 shift right.
%e Binomial transform of row 3 forms column 3 of square A100324: BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].
%e Binomial transform of row 4 forms column 4 of square A100324: BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].
%p A100326 := proc(n,k)
%p if k < 0 or k > n then
%p 0 ;
%p elif n = 0 then
%p 1 ;
%p elif k = 0 then
%p A003169(n)
%p else
%p add(procname(i+1,0)*procname(n-i-1,k-1),i=0..n-k) ;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 15 2013
%t 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 *)
%o (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)));))
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (Haskell)
%o import Data.List (transpose)
%o a100326 n k = a100326_tabl !! n !! k
%o a100326_row n = a100326_tabl !! n
%o a100326_tabl = [1] : f [[1]] where
%o f xss@(xs:_) = ys : f (ys : xss) where
%o ys = y : map (sum . zipWith (*) (zs ++ [y])) (map reverse zss)
%o y = sum $ zipWith (*) [1..] xs
%o zss@((_:zs):_) = transpose $ reverse xss
%o -- _Reinhard Zumkeller_, Nov 21 2015
%o (SageMath)
%o @CachedFunction
%o def T(n,k): # T = A100326
%o if (k<0 or k>n): return 0
%o elif (k==n): return 1
%o elif (k==0): return sum((j+1)*T(n-1,j) for j in range(n))
%o else: return sum(T(j+1,0)*T(n-j-1,k-1) for j in range(n-k+1))
%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 30 2023
%Y Cf. A003169, A100324, A100327 (row sums), A158687, A264717 (central terms).
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Nov 17 2004
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