cp's OEIS Frontend

A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).

%I A108350 #9 Dec 05 2016 02:40:41
%S A108350 1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,13,13,5,1,1,6,21,32,21,6,1,1,7,31,
%T A108350 65,65,31,7,1,1,8,43,116,161,116,43,8,1,1,9,57,189,341,341,189,57,9,1,
%U A108350 1,10,73,288,645,842,645,288,73,10,1,1,11,91,417,1121,1827,1827,1121,417,91
%N A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).
%C A108350 Or as a square array read by antidiagonals, T(n,k) = Sum_{j=0..n} binomial(k,j)*binomial(n+k-j,k)*((j+1) mod 2).
%C A108350 A symmetric number triangle based on 1/(1-x^2).
%C A108350 The construction of a symmetric triangle in this example is general. Let f(n) be a sequence, preferably with f(0)=1. Then T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*f(j) yields a symmetric triangle. When f(n)=1^n, we get Pascal's triangle. When f(n)=2^n, we get the Delannoy triangle (see A008288). In general, f(n)=k^n yields a (1,k,1)-Pascal triangle (see A081577, A081578). Row sums of triangle are A100131. Diagonal sums of the triangle are A108351. Triangle mod 2 is A106465.
%F A108350 Row k (and column k) has g.f. (1+C(k,2)x^2)/(1-x)^(k+1).
%e A108350 Triangle rows begin
%e A108350   1;
%e A108350   1,  1;
%e A108350   1,  2,  1;
%e A108350   1,  3,  3,  1;
%e A108350   1,  4,  7,  4,  1;
%e A108350   1,  5, 13, 13,  5,  1;
%e A108350   1,  6, 21, 32, 21,  6,  1;
%e A108350 As a square array read by antidiagonals, rows begin
%e A108350   1, 1,  1,   1,   1,    1,    1, ...
%e A108350   1, 2,  3,   4,   5,    6,    7, ...
%e A108350   1, 3,  7,  13,  21,   31,   43, ...
%e A108350   1, 4, 13,  32,  65,  116,  189, ...
%e A108350   1, 5, 21,  65, 161,  341,  645, ...
%e A108350   1, 6, 31, 116, 341,  842, 1827, ...
%e A108350   1, 7, 43, 189, 645, 1827, 4495, ...
%o A108350 (PARI) trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ _Michel Marcus_, Sep 11 2013
%K A108350 easy,nonn,tabl
%O A108350 0,5
%A A108350 _Paul Barry_, May 31 2005