%I #12 Nov 04 2024 12:22:43

%S 1,5,21,92,414,1869,8427,37975,171121,771119,3474913,15659094,

%T 70564951,317988473,1432958824,6457375642,29099021980,131129599227,

%U 590912361256,2662842109828,11999627299824,54074199444301,243675821963849,1098082020999797,4948312537227216

%N Number of compositions of 6*n-3 into parts 1 and 6.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,20,-15,6,-1).

%F a(n) = A005708(6*n-3).

%F a(n) = Sum_{k=0..n} binomial(n+2+5*k,n-1-k).

%F a(n) = 7*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

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

%F a(n) = A373302(n) - A373302(n-1).

%t LinearRecurrence[{7,-15,20,-15,6,-1},{1,5,21,92,414,1869},30] (* _Harvey P. Dale_, Nov 04 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n+2+5*k, n-1-k));

%Y Cf. A099242, A371125, A373302, A373959, A373960.

%Y Cf. A005708.

%K nonn,easy,changed

%O 1,2

%A _Seiichi Manyama_, Jun 23 2024