cp's OEIS Frontend

A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.

%I A167772 #19 May 27 2022 08:09:23
%S A167772 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 A167772 243,186,111,54,21,6,1,622,808,622,379,193,82,28,7,1,2120,2742,2120,
%U A167772 1312,690,311,118,36,8,1,7338,9458,7338,4596,2476,1164,474,163,45,9,1
%N A167772 Riordan array (c(x)/(1+x*c(x)), x*c(x)), c(x) the g.f. of A000108.
%H A167772 Reinhard Zumkeller, <a href="/A167772/b167772.txt">Rows n=0..125 of triangle, flattened</a>
%F A167772 Sum_{k=0..n} T(n, k) = A000958(n+1).
%F A167772 From _Philippe Deléham_, Nov 12 2009: (Start)
%F A167772 Sum_{k=0..n} T(n,k)*2^k = A014300(n).
%F A167772 Sum_{k=0..n} T(n,k)*2^(n-k) = A064306(n). (End)
%F A167772 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 A167772 Triangle begins:
%e A167772      1;
%e A167772      0,    1;
%e A167772      1,    1,    1;
%e A167772      2,    3,    2,    1;
%e A167772      6,    8,    6,    3,   1;
%e A167772     18,   24,   18,   10,   4,   1;
%e A167772     57,   75,   57,   33,  15,   5,   1;
%e A167772    186,  243,  186,  111,  54,  21,   6,  1;
%e A167772    622,  808,  622,  379, 193,  82,  28,  7,  1;
%e A167772   2120, 2742, 2120, 1312, 690, 311, 118, 36,  8,  1;
%e A167772 Production matrix begins:
%e A167772   0, 1;
%e A167772   1, 1, 1;
%e A167772   1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A167772   1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A167772   ... - _Philippe Deléham_, Mar 03 2013
%t A167772 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 A167772 T[n_, k_]:= If[k==0, A065602[n+1,3] + Boole[n==0], A065602[n+1, k+1]];
%t A167772 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 26 2022 *)
%o A167772 (Haskell)
%o A167772 import Data.List (genericIndex)
%o A167772 a167772 n k = genericIndex (a167772_row n) k
%o A167772 a167772_row n = genericIndex a167772_tabl n
%o A167772 a167772_tabl = [1] : [0, 1] :
%o A167772                map (\xs@(_:x:_) -> x : xs) (tail a065602_tabl)
%o A167772 -- _Reinhard Zumkeller_, May 15 2014
%o A167772 (SageMath)
%o A167772 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 A167772 def A167772(n,k):
%o A167772     if (k==0): return A065602(n+1,3) + bool(n==0)
%o A167772     else: return A065602(n+1,k+1)
%o A167772 flatten([[A167772(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2022
%Y A167772 Cf. A000957, A000958 (row sums), A001558, A001559, A104629, A167769.
%Y A167772 Diagonals: A000012, A000217, A001477, A166830.
%K A167772 nonn,tabl
%O A167772 0,7
%A A167772 _Philippe Deléham_, Nov 11 2009, corrected Nov 12 2009