A167772 - OEIS

%I #19 May 27 2022 08:09:23

%S 1,0,1,1,1,1,2,3,2,1,6,8,6,3,1,18,24,18,10,4,1,57,75,57,33,15,5,1,186,

%T 243,186,111,54,21,6,1,622,808,622,379,193,82,28,7,1,2120,2742,2120,

%U 1312,690,311,118,36,8,1,7338,9458,7338,4596,2476,1164,474,163,45,9,1

%N Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.

%H Reinhard Zumkeller, <a href="/A167772/b167772.txt">Rows n=0..125 of triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A000958(n+1).

%F From _Philippe Deléham_, Nov 12 2009: (Start)

%F Sum_{k=0..n} T(n,k)*2^k = A014300(n).

%F Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)

%F For n > 0: T(n,0) = A065602(n+1,3), T(n,k) = A065602(n+1,k+1), k = 1..n. - _Reinhard Zumkeller_, May 15 2014

%e Triangle begins:

%e      1;

%e      0,    1;

%e      1,    1,    1;

%e      2,    3,    2,    1;

%e      6,    8,    6,    3,   1;

%e     18,   24,   18,   10,   4,   1;

%e     57,   75,   57,   33,  15,   5,   1;

%e    186,  243,  186,  111,  54,  21,   6,  1;

%e    622,  808,  622,  379, 193,  82,  28,  7,  1;

%e   2120, 2742, 2120, 1312, 690, 311, 118, 36,  8,  1;

%e Production matrix begins:

%e   0, 1;

%e   1, 1, 1;

%e   1, 1, 1, 1;

%e   1, 1, 1, 1, 1;

%e   1, 1, 1, 1, 1, 1;

%e   1, 1, 1, 1, 1, 1, 1;

%e   1, 1, 1, 1, 1, 1, 1, 1;

%e   1, 1, 1, 1, 1, 1, 1, 1, 1;

%e   1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

%e   ... - _Philippe Deléham_, Mar 03 2013

%t A065602[n_, k_]:= A065602[n,k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n-j) -k-1), {j, 0, (n-k)/2}];

%t T[n_, k_]:= If[k==0, A065602[n+1,3] + Boole[n==0], A065602[n+1, k+1]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 26 2022 *)

%o (Haskell)

%o import Data.List (genericIndex)

%o a167772 n k = genericIndex (a167772_row n) k

%o a167772_row n = genericIndex a167772_tabl n

%o a167772_tabl = [1] : [0, 1] :

%o                map (\xs@(_:x:_) -> x : xs) (tail a065602_tabl)

%o -- _Reinhard Zumkeller_, May 15 2014

%o (SageMath)

%o def A065602(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )

%o def A167772(n,k):

%o     if (k==0): return A065602(n+1,3) + bool(n==0)

%o     else: return A065602(n+1,k+1)

%o flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2022

%Y Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.

%Y Diagonals: A000012, A000217, A001477, A166830.

%K nonn,tabl

%O 0,7

%A _Philippe Deléham_, Nov 11 2009, corrected Nov 12 2009