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A114172 Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1), for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1).

%I A114172 #5 Nov 23 2021 05:59:52
%S A114172 1,1,1,1,3,1,1,5,6,1,1,7,16,9,1,1,9,31,36,12,1,1,11,51,95,66,15,1,1,
%T A114172 13,76,199,229,106,18,1,1,15,106,361,601,467,156,21,1,1,17,141,594,
%U A114172 1316,1509,844,216,24,1,1,19,181,911,2542,3951,3293,1395,286,27,1,1,21,226
%N A114172 Triangle, read by rows, where the g.f. of column n, C_n(x), equals the g.f. of row n, R_n(x), divided by (1-x)^(n+1), for n>=0; e.g., C_n(x) = R_n(x)/(1-x)^(n+1).
%e A114172 Triangle begins:
%e A114172 1;
%e A114172 1,1;
%e A114172 1,3,1;
%e A114172 1,5,6,1;
%e A114172 1,7,16,9,1;
%e A114172 1,9,31,36,12,1;
%e A114172 1,11,51,95,66,15,1;
%e A114172 1,13,76,199,229,106,18,1;
%e A114172 1,15,106,361,601,467,156,21,1;
%e A114172 1,17,141,594,1316,1509,844,216,24,1;
%e A114172 1,19,181,911,2542,3951,3293,1395,286,27,1;
%e A114172 1,21,226,1325,4481,8910,10193,6447,2155,366,30,1; ...
%e A114172 Where g.f. for columns is formed from g.f. of rows:
%e A114172 GF(column 2) = (1 + 3*x + 1*x^2)/(1-x)^3
%e A114172 = 1 + 6*x + 16*x^2 + 31*x^3 + 51*x^4 + 76*x^5 +...
%e A114172 GF(column 3) = (1 + 5*x + 6*x^2 + 1*x^3)/(1-x)^4
%e A114172 = 1 + 9*x + 36*x^2 + 95*x^3 + 199*x^4 + 361*x^5 +...
%e A114172 GF(column 4) = (1 + 7*x + 16*x^2 + 9*x^3 + 1*x^4)/(1-x)^5
%e A114172 = 1 + 12*x + 66*x^2 + 229*x^3 + 601*x^4 + 1316*x^5 +...
%o A114172 (PARI)
%o A114172 {T(n,k)=if(n<k||k<0,0,if(n==k||k==0,1, polcoeff(sum(j=0,k,T(k,j)*x^j)/(1-x+x*O(x^(n-k)))^(k+1),n-k)))}
%Y A114172 Cf. A114173 (row sums), A114174 (central terms), A114175 (row sums-square).
%K A114172 nonn,tabl
%O A114172 0,5
%A A114172 _Paul D. Hanna_, Nov 15 2005