A190090 - OEIS

%I #26 Sep 08 2022 08:45:56

%S 1,1,4,16,42,137,443,1365,4316,13625,42785,134758,424331,1335378,

%T 4203927,13233947,41657808,131135696,412803240,1299458257,4090567673,

%U 12876698159,40534529294,127598621869,401667591501,1264408966284,3980231826575,12529367967276,39441185140197

%N Diagonal sums of the triangular matrix A190088.

%H Vincenzo Librandi, <a href="/A190090/b190090.txt">Table of n, a(n) for n = 0..201</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,6,-3,0,1).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k+1,3*n-6*k+1).

%F G.f.: (1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6).

%F a(n) = 2*a(n-1)+ 2*a(n-2)+ 6*a(n-3)-3*a(n-4)+a(n-6), and a(0)=1, a(1)=1, a(2)=4, a(3)=16, a(4)=42, a(5)=137, . - _Harvey P. Dale_, Jul 04 2011

%t Table[Sum[Binomial[3n - 4k + 1, 3n - 6k + 1], {k, 0, n/2}], {n, 0, 26}]

%t LinearRecurrence[{2,2,6,-3,0,1},{1,1,4,16,42,137},27] (* _Harvey P. Dale_, Jul 04 2011 *)

%o (Maxima) makelist(sum(binomial(3*n-4*k+1,3*n-6*k+1),k,0,n/2),n,0,12);

%o (PARI) Vec((1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6)+O(x^29)) \\ _Charles R Greathouse IV_, Jun 30 2011

%o (Magma) [(&+[Binomial(3*n-4*k+1,3*n-6*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Mar 04 2018

%Y Cf. A190088, A190089.

%K nonn,easy

%O 0,3

%A _Emanuele Munarini_, May 04 2011