A147703 - OEIS

%I #27 May 28 2021 07:58:37

%S 1,1,1,2,3,1,5,9,5,1,13,27,20,7,1,34,80,73,35,9,1,89,234,252,151,54,

%T 11,1,233,677,837,597,269,77,13,1,610,1941,2702,2225,1199,435,104,15,

%U 1,1597,5523,8533,7943,4956,2158,657,135,17,1

%N Triangle [1,1,1,0,0,0,...] DELTA [1,0,0,0,...] with Deléham DELTA defined in A084938.

%C Equal to A062110*A007318 when A062110 is regarded as a triangle read by rows.

%H Indranil Ghosh, <a href="/A147703/b147703.txt">Rows 0..100, flattened</a>

%F Riordan array ((1-2x)/(1-3x+x^2), x(1-x)/(1-3x+x^2)).

%F Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - _Philippe Deléham_, Dec 01 2008

%F G.f.: (1-2*x)/(1-(3+y)*x+(1+y)*x^2). - _Philippe Deléham_, Nov 26 2011

%F T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), for n > 1. - _Philippe Deléham_, Feb 12 2012

%F The Riordan square of the odd indexed Fibonacci numbers A001519. - _Peter Luschny_, Jan 26 2020

%e Triangle begins

%e    1;

%e    1,   1;

%e    2,   3,   1;

%e    5,   9,   5,   1;

%e   13,  27,  20,   7,  1;

%e   34,  80,  73,  35,  9,  1;

%e   89, 234, 252, 151, 54, 11, 1;

%p # The function RiordanSquare is defined in A321620:

%p RiordanSquare(1 / (1 - x / (1 - x / (1 - x))), 10); # _Peter Luschny_, Jan 26 2020

%t nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - 2*x)/(1 - (3 + y)*x + (1 + y)*x^2), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* _Indranil Ghosh_, Mar 11 2017 *)

%Y Row sums are A006012. Diagonal sums are A147704.

%Y Cf. A062110, A007318, A206800, A001519, A321620.

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Nov 10 2008