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Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.
15

%I #25 Sep 08 2022 08:45:30

%S 1,2,4,7,11,16,23,37,74,175,431,1024,2291,4825,9650,18571,34955,65536,

%T 124511,242461,484922,989527,2038103,4194304,8565755,17308657,

%U 34617314,68703187,135812051,268435456,532087943,1059392917,2118785834

%N Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

%H Vincenzo Librandi, <a href="/A129339/b129339.txt">Table of n, a(n) for n = 1..1000</a>

%H Paul Curtz, <a href="/A129339/a129339.txt">Comments on this sequence</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,6).

%F G.f.: x*(1-x)^3/((1-2*x)*(1-3*x+3*x^2)). [multiplied by x to match the offset by _R. J. Mathar_, Jul 22 2009]

%F a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 7; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).

%F Binomial transform of A088911. - _Klaus Brockhaus_, Jun 17 2007

%F a(n+1) = A057083(n)/3+2^(n-1), n > 1. - _R. J. Mathar_, Jul 22 2009

%e First seven rows of T are

%e [ 1 ]

%e [ 1, 2 ]

%e [ 1, 2, 4 ]

%e [ 0, 1, 3, 7 ]

%e [ 0, 0, 1, 4, 11 ]

%e [ 0, 0, 0, 1, 5, 16 ]

%e [ 1, 1, 1, 1, 2, 7, 23 ].

%t a[n_] := 2^(n-2) + 2*3^((n-3)/2)*Sin[n*Pi/6]; a[1]=1; Table[a[n], {n, 1, 33}] (* _Jean-François Alcover_, Aug 13 2012 *)

%t CoefficientList[Series[(1 - x)^3 / ((1 - 2 x) (1 - 3 x + 3 x^2)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 13 2018 *)

%o (PARI) {m=33; v=concat([1, 2, 4, 7], vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v} \\ _Klaus Brockhaus_, Jun 10 2007

%o (Magma) m:=33; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n, n]: n in [1..m] ]; // _Klaus Brockhaus_, Jun 10 2007

%o (Magma) m:=33; S:=[ [1, 1, 1, 0, 0, 0][(n-1) mod 6 + 1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // _Klaus Brockhaus_, Jun 17 2007

%o (Magma) I:=[1,2,4,7]; [n le 4 select I[n] else 5*Self(n-1)-9*Self(n-2)+6*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 13 2018

%Y Cf. A038504, A131022 (T read by rows), A131023 (first subdiagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

%K nonn,easy

%O 1,2

%A _Paul Curtz_, May 28 2007

%E Edited and extended by _Klaus Brockhaus_, Jun 10 2007