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%I #19 Feb 26 2022 09:20:33
%S 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 30,20,7,1,0,2,13,36,55,50,27,8,1,0,2,15,49,91,105,77,35,9,1
%N A129686 * A007318. Riordan triangle (1+x, x/(1-x)).
%C Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48, ...). A131085 = A007318 * A129686
%C Riordan array (1+x, x/(1-x)). - _Philippe Deléham_, Mar 02 2012
%F 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 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 G.f. column k: (1+x)*(x/(1-x))^k, k >= 0. (Riordan property). - _Wolfdieter Lang_, Jan 06 2015
%F 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 The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 1 1
%e 2: 0 2 1
%e 3: 0 2 3 1
%e 4: 0 2 5 4 1
%e 5: 0 2 7 9 5 1
%e 6: 0 2 9 16 14 6 1
%e 7: 0 2 11 25 30 20 7 1
%e 8: 0 2 13 36 55 50 27 8 1
%e 9: 0 2 15 49 91 105 77 35 9 1
%e 10: 0 2 17 64 140 196 182 112 44 10 1
%e ... Reformatted. - _Wolfdieter Lang_, Jan 06 2015
%Y Cf. A007318, A129686, A098011, A131085.
%Y Cf. A029653, A131084, A208510.
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, Jun 14 2007
%E Edited: Added Riordan property (see Philippe Deléham comment) in name. - _Wolfdieter Lang_, Jan 06 2015