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A378145 Riordan triangle (1 + x * C(x), x * C(x)), where C(x) is g.f. of A000108.

%I A378145 #11 Dec 08 2024 17:12:07
%S A378145 1,1,1,1,2,1,2,4,3,1,5,10,8,4,1,14,28,23,13,5,1,42,84,70,42,19,6,1,
%T A378145 132,264,222,138,68,26,7,1,429,858,726,462,240,102,34,8,1,1430,2860,
%U A378145 2431,1573,847,385,145,43,9,1,4862,9724,8294,5434,3003,1430,583,198,53,10,1
%N A378145 Riordan triangle (1 + x * C(x), x * C(x)), where C(x) is g.f. of A000108.
%F A378145 T(n, k) = binomial(2*n-k, n) * (n*(3*k+1) - 2*k*(k+1)) / ((2*n-k) * (2*n-k-1)) if 0 <= k < n and 1 if k = n.
%F A378145 T(n, k) = T(n, k-1) - T(n-1, k-2) for 2 <= k <= n.
%F A378145 (-1)^(n-k) * T(n, k) is matrix inverse of A004070 (seen as a triangle).
%F A378145 Conjecture: Sum_{i=0..n-k} binomial(i+m-1, i) * T(n, i+k) = T(n+m, m+k) for m > 0.
%F A378145 Conjecture: Sum_{k=0..n} (1 + floor(k/2)) * T(n, k) = A000108(n+1).
%F A378145 G.f.: A(x, y) = (1 + x*C(x)) / (1 - y * x*C(x)), where C(x) is g.f. of A000108.
%e A378145 Triangle T(n, k) for 0 <= k <= n starts:
%e A378145 n\k :     0     1     2     3    4    5    6   7  8  9
%e A378145 ======================================================
%e A378145   0 :     1
%e A378145   1 :     1     1
%e A378145   2 :     1     2     1
%e A378145   3 :     2     4     3     1
%e A378145   4 :     5    10     8     4    1
%e A378145   5 :    14    28    23    13    5    1
%e A378145   6 :    42    84    70    42   19    6    1
%e A378145   7 :   132   264   222   138   68   26    7   1
%e A378145   8 :   429   858   726   462  240  102   34   8  1
%e A378145   9 :  1430  2860  2431  1573  847  385  145  43  9  1
%e A378145   etc.
%o A378145 (PARI) T(n,k)=if(k==n,1,binomial(2*n-k,n)*(n*(3*k+1)-2*k*(k+1))/((2*n-k)*(2*n-k-1)))
%Y A378145 Cf. A000108, A004070, A120588 (column 0), A068875 (column 1 and row sums), A000007 (alt. row sums).
%K A378145 nonn,easy,tabl
%O A378145 0,5
%A A378145 _Werner Schulte_, Nov 17 2024