This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374454 #19 Aug 20 2024 22:00:16
%S A374454 1,4,22,116,613,3240,17124,90504,478333,2528092,13361506,70618412,
%T A374454 373233385,1972618128,10425707976,55102092624,291226324249,
%U A374454 1539193302772,8134965235054,42995028146468,227237903531533,1201000837247928,6347545848001836,33548135057767512
%N A374454 Expansion of o.g.f. 1/(1 - 4*x - 6*x^2 - 4*x^3 - x^4).
%C A374454 a(n) is the number of generalized compositions of n using parts of size at most 4 where there are binomial(4,i) types of i (see example).
%C A374454 The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i.
%C A374454 Related sequences that count the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i are A108368(n+1), A000129(n+1), and A000012(n) for k = 3, 2, 1, respectively.
%H A374454 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,6,4,1).
%F A374454 a(n) = 4*a(n-1) + 6*a(n-2) + 4*a(n-3) + a(n-4), n=>4.
%F A374454 a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(4*k,n). - _Seiichi Manyama_, Aug 03 2024
%e A374454 The following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 6 using parts of size at most 4 where there are binomial(4,i) types of i (ie. 4 types of 1, 6 types of 2, 4 types of 3 and 1 type of 4):
%e A374454 Type Number Total
%e A374454 4+2 2 12
%e A374454 3+3 1 16
%e A374454 4+1+1 3 48
%e A374454 3+2+1 6 576
%e A374454 2+2+2 1 216
%e A374454 3+1+1+1 4 1024
%e A374454 2+2+1+1 6 3456
%e A374454 2+1+1+1+1 5 7680
%e A374454 1+1+1+1+1+1 1 4096,
%e A374454 adding to a(6) = 17124.
%t A374454 CoefficientList[Series[1/(1-4*x-6*x^2-4*x^3-x^4),{x,0,23}],x] (* _Stefano Spezia_, Jul 09 2024 *)
%Y A374454 Cf. A000012, A000129, A108368, A374455.
%Y A374454 Cf. A145840.
%K A374454 nonn,easy
%O A374454 0,2
%A A374454 _Enrique Navarrete_, Jul 08 2024