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 A368475 #43 Feb 02 2024 23:56:20
%S A368475 1,0,0,0,1,5,15,35,71,136,265,550,1211,2732,6126,13485,29191,62648,
%T A368475 134408,289656,627401,1363124,2963186,6434484,13951852,30221185,
%U A368475 65442625,141745045,307137901,665732417,1443184210,3128438335,6780867186,14696002913,31848721632
%N A368475 Expansion of o.g.f. (1-x)^5/((1-x)^5 - x^4).
%C A368475 For n > 0, a(n) is the number of ways to split [n] into an unspecified number of intervals and then choose 4 blocks (i.e., subintervals) from each interval. For example, for n=12, a(12)=1211 since the number of ways to split [12] into intervals and then select 4 blocks from each interval is C(12,4) + C(8,4)*C(4,4) + C(7,4)*C(5,4) + C(6,4)*C(6,4) + C(5,4)*C(7,4) + C(4,4)*C(8,4) + C(4,4)*C(4,4)*C(4,4) for a total of 1211 ways.
%C A368475 For n > 0, a(n) is also the number of compositions of n using parts of size at least 4 where there are binomial(i,4) types of i, i >= 4 (see example).
%C A368475 Number of compositions of 5*n-4 into parts 4 and 5. - _Seiichi Manyama_, Feb 01 2024
%H A368475 Seiichi Manyama, <a href="/A368475/b368475.txt">Table of n, a(n) for n = 0..1000</a>
%H A368475 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-4,1).
%F A368475 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 4*a(n-4) + a(n-5), n>=6; a(0)=1, a(1)=a(2)=a(3)=0, a(4)=1, a(5)=5.
%F A368475 G.f.: 1/(1-Sum_{k>=4} binomial(k,4)*x^k).
%F A368475 G.f.: 1/p(S), where p(S) = 1 - S^4 - S^5 and S = x/(1-x).
%F A368475 First differences of A099131. - _R. J. Mathar_, Jan 29 2024
%F A368475 a(n) = A017827(5*n-4) = Sum_{k=0..floor((5*n-4)/4)} binomial(k,5*n-4-4*k) for n > 0. - _Seiichi Manyama_, Feb 01 2024
%F A368475 a(n) = Sum_{k=0..floor(n/4)} binomial(n-1+k,n-4*k). - _Seiichi Manyama_, Feb 02 2024
%e A368475 Since there are C(4,4) = 1 type of 4, C(5,4) = 5 types of 5, C(6,4) = 15 types of 6, C(7,4) = 35 types of 7, C(8,4) = 70 types of 8, and (12,4) = 495 types of 12, we can write 12 in the following ways:
%e A368475 12: 495 ways;
%e A368475 8+4: 70 ways;
%e A368475 7+5: 175 ways;
%e A368475 6+6: 225 ways;
%e A368475 5+7: 175 ways;
%e A368475 4+8: 70 ways;
%e A368475 4+4+4: 1 way, for a total of 1211 ways.
%t A368475 CoefficientList[Series[(1 - x)^5/((1 - x)^5 - x^4), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Dec 26 2023 *)
%o A368475 (PARI) Vec((1-x)^5/((1-x)^5 - x^4) + O(x^40)) \\ _Michel Marcus_, Dec 27 2023
%Y A368475 Cf. A000332, A017827, A099131, A290998.
%Y A368475 Cf. A079675, A369803, A369804.
%Y A368475 Cf. A088305, A095263, A290998, A369794, A369809.
%K A368475 nonn,easy
%O A368475 0,6
%A A368475 _Enrique Navarrete_, Dec 26 2023