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A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).

%I A131084 #19 Feb 26 2022 09:20:33
%S A131084 1,1,1,0,2,1,0,2,3,1,0,2,5,4,1,0,2,7,9,5,1,0,2,9,16,14,6,1,0,2,11,25,
%T A131084 30,20,7,1,0,2,13,36,55,50,27,8,1,0,2,15,49,91,105,77,35,9,1
%N A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).
%C A131084 Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48, ...). A131085 = A007318 * A129686
%C A131084 Riordan array (1+x, x/(1-x)). - _Philippe Deléham_, Mar 02 2012
%F A131084 A129686(signed): (1,1,1,...) in the main diagonal and (-1,-1,-1, ...) in the subsubdiagonal); * A007318, Pascal's triangle; as infinite lower triangular matrices.
%F A131084 exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2*x + 3*x^2/2! + x^3/3!) = 2*x + 7*x^2/2! + 16*x^3/3! + 30*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - _Peter Bala_, Dec 22 2014
%F A131084 G.f. column k: (1+x)*(x/(1-x))^k, k >= 0. (Riordan property). - _Wolfdieter Lang_, Jan 06 2015
%F A131084 T(n, 0) = 1 if n=0 or n=1 else 0; T(n, k) = binomial(n-1,k-1) + binomial(n-2,k-1)*[n-1 >= k] if n >= k >= 1, where [S] = 1 if S is true, else 0, and  T(n, k) = 0 if n < k. - _Wolfdieter Lang_, Jan 08 2015
%e A131084 The triangle T(n, k) begins:
%e A131084 n\k 0  1  2  3   4   5   6   7  8  9 10 ...
%e A131084 0:  1
%e A131084 1:  1  1
%e A131084 2:  0  2  1
%e A131084 3:  0  2  3  1
%e A131084 4:  0  2  5  4   1
%e A131084 5:  0  2  7  9   5   1
%e A131084 6:  0  2  9 16  14   6   1
%e A131084 7:  0  2 11 25  30  20   7   1
%e A131084 8:  0  2 13 36  55  50  27   8  1
%e A131084 9:  0  2 15 49  91 105  77  35  9  1
%e A131084 10: 0  2 17 64 140 196 182 112 44 10  1
%e A131084 ... Reformatted. - _Wolfdieter Lang_, Jan 06 2015
%Y A131084 Cf. A007318, A129686, A098011, A131085.
%Y A131084 Cf. A029653, A131084, A208510.
%K A131084 nonn,tabl
%O A131084 1,5
%A A131084 _Gary W. Adamson_, Jun 14 2007
%E A131084 Edited: Added Riordan property (see Philippe Deléham comment) in name. - _Wolfdieter Lang_, Jan 06 2015