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 A351292 #15 Feb 16 2022 15:40:40
%S A351292 1,1,1,5,5,9,57,61,109,161,1265,1317,2469,3577,5785,43901,47165,86337,
%T A351292 127665,204853,284197,2280089,2398505,4469373,6543453,10570993,
%U A351292 14601745,22502549,159506453,171281529,314077353,462623821,742191037,1031307185,1580543969,2141246229
%N A351292 Number of patterns of length n with all distinct run-lengths.
%C A351292 We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
%H A351292 Andrew Howroyd, <a href="/A351292/b351292.txt">Table of n, a(n) for n = 0..1000</a>
%F A351292 From _Andrew Howroyd_, Feb 12 2022: (Start)
%F A351292 a(n) = Sum_{k=1..n} R(n,k)*(Sum_{r=k..n} binomial(r, k)*(-1)^(r-k)), where R(n,k) = Sum_{j=1..floor((sqrt(8*n+1)-1)/2)} k*(k-1)^(j-1) * j! * A008289(n,j).
%F A351292 G.f.: 1 + Sum_{r>=1} Sum_{k=1..r} R(k,x) * binomial(r, k)*(-1)^(r-k), where R(k,x) = Sum_{j>=1} k*(k-1)^(j-1) * j! * [y^j](Product_{k>=1} 1 + y*x^k).
%F A351292 (End)
%e A351292 The a(1) = 1 through a(5) = 9 patterns:
%e A351292 (1) (1,1) (1,1,1) (1,1,1,1) (1,1,1,1,1)
%e A351292 (1,1,2) (1,1,1,2) (1,1,1,1,2)
%e A351292 (1,2,2) (1,2,2,2) (1,1,1,2,2)
%e A351292 (2,1,1) (2,1,1,1) (1,1,2,2,2)
%e A351292 (2,2,1) (2,2,2,1) (1,2,2,2,2)
%e A351292 (2,1,1,1,1)
%e A351292 (2,2,1,1,1)
%e A351292 (2,2,2,1,1)
%e A351292 (2,2,2,2,1)
%e A351292 The a(6) = 57 patterns grouped by sum:
%e A351292 111111 111112 111122 112221 111223 111233 112333 122333
%e A351292 111211 111221 122211 111322 111332 113332 133322
%e A351292 112111 122111 211122 112222 112223 122233 221333
%e A351292 211111 221111 221112 211222 113222 133222 223331
%e A351292 221113 122222 211333 333122
%e A351292 222112 211133 222133 333221
%e A351292 222211 221222 222331
%e A351292 223111 222113 233311
%e A351292 311122 222122 331222
%e A351292 322111 222221 332221
%e A351292 222311 333112
%e A351292 233111 333211
%e A351292 311222
%e A351292 322211
%e A351292 331112
%e A351292 332111
%t A351292 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A351292 Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Length/@Split[#]&]],{n,0,6}]
%o A351292 (PARI)
%o A351292 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A351292 R(u,k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A351292 seq(n)={my(u=P(n), c=poldegree(u[#u])); concat([1], sum(k=1, c, R(u, k)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)) ))} \\ _Andrew Howroyd_, Feb 11 2022
%Y A351292 The version for runs instead of run-lengths is A351200.
%Y A351292 A000670 counts patterns, ranked by A333217.
%Y A351292 A005649 counts anti-run patterns, complement A069321.
%Y A351292 A005811 counts runs in binary expansion.
%Y A351292 A032011 counts patterns with distinct multiplicities.
%Y A351292 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351292 A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
%Y A351292 A131689 counts patterns by number of distinct parts.
%Y A351292 A238130 and A238279 count compositions by number of runs.
%Y A351292 A165413 counts distinct run-lengths in binary expansion, runs A297770.
%Y A351292 A345194 counts alternating patterns, up/down A350354.
%Y A351292 Counting words with all distinct runs:
%Y A351292 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351292 - A351016 = binary words, for run-lengths A351017.
%Y A351292 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351292 - A351202 = permutations of prime factors.
%Y A351292 - A351638 = word structures.
%Y A351292 Row sums of A350824.
%Y A351292 Cf. A003242, A098504, A098859, A106356, A239455, A242882, A325545, A328592, A329740, A351014, A351293.
%K A351292 nonn
%O A351292 0,4
%A A351292 _Gus Wiseman_, Feb 10 2022
%E A351292 Terms a(10) and beyond from _Andrew Howroyd_, Feb 11 2022