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 A261319 #12 Aug 15 2015 06:58:02
%S A261319 1,0,0,0,1,2,0,0,3,4,0,1,11,19,20,0,0,30,80,95,96,0,1,92,372,527,551,
%T A261319 552,0,0,273,1764,3129,3500,3535,3536,0,1,821,8549,19595,24299,25055,
%U A261319 25103,25104
%N A261319 Number of set partitions C'_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks and such that no part contains both 1 and t (each unmarked) or both i and i+1 (each unmarked) for some i with 1 <= i < t; triangle C'_t(n), t>=0, 0<=n<=t, read by rows.
%C A261319 C'_t(n) is the number of sequences of t non-identity top-to-random shuffles that leave a deck of n cards invariant, if each shuffle is permitted to flip the orientation of the card it moves.
%C A261319 C'_t(n) = <(pi-1_{BSym_n})^t, 1_{BSym_n}> where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C'_t(n) using sequences of box moves on pairs of Young diagrams.
%C A261319 C'_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks and such that no part contains both 1 and t (each unmarked) or both i and i+1 (each unmarked) for some i with 1 <= i < t.
%C A261319 C'_t(n) = C'_t(t) if n > t.
%H A261319 John R. Britnell and Mark Wildon, <a href="http://arxiv.org/abs/1507.04803">Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D</a>, arXiv:1507.04803 [math.CO], 2015.
%F A261319 C'_t(n) + C'_t(n-1) = Sum_{s=0..t-1} binomial(t-1,s)*A261275(s,n-1) for n>=1.
%F A261319 E.g.f.: diagonal is exp(1/2*(exp(2*x)-2*x-1)).
%F A261319 C'_t(n) = Sum_{i=0..n} A261318(t,i).
%e A261319 Triangle starts:
%e A261319 1;
%e A261319 0, 0;
%e A261319 0, 1, 2;
%e A261319 0, 0, 3, 4;
%e A261319 0, 1, 11, 19, 20;
%e A261319 0, 0, 30, 80, 95, 96;
%e A261319 0, 1, 92, 372, 527, 551, 552;
%e A261319 0, 0, 273, 1764, 3129, 3500, 3535, 3536;
%e A261319 0, 1, 821, 8549, 19595, 24299, 25055, 25103, 25104;
%t A261319 TGF[1, x_] := x^2/(1 - x^2); TGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - (2*j - 1)*x), {j, 1, n}];
%t A261319 T[0, 0] := 1; T[_, 0] := 0; T[0, _] := 0; T[t_, n_] := Coefficient[Series[TGF[n, x], {x, 0, t}], x^t];
%t A261319 CC[t_, n_] := Sum[T[t, m], {m, 0, n}]
%Y A261319 Cf. A261275, A261318, A261139, A261137.
%K A261319 nonn,tabl
%O A261319 0,6
%A A261319 _Mark Wildon_, Aug 14 2015