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 A132892 #24 Jan 28 2025 08:33:18
%S A132892 1,1,1,1,2,1,1,3,3,1,1,4,5,3,1,1,5,9,7,4,1,1,6,13,14,10,4,1,1,7,19,25,
%T A132892 22,12,5,1,1,8,25,41,42,30,15,5,1,1,9,33,63,79,66,43,19,6,1,1,10,41,
%U A132892 92,131,132,99,55,22,6,1,1,11,51,129,213,245,217,143,73,26,7,1,1,12,61,175,325,428,429,335,201,91,31,7,1
%N A132892 Square array T(m,n) read by antidiagonals; T(m,n) is the number of equivalence classes in the set of sequences of n nonnegative integers that sum to m, generated by the equivalence relation defined in the following manner: we write a sequence in the form a[1]0a[2]0...0a[p], where each a[i] is a (possibly empty) sequence of positive integers; two sequences in this form, a[1]0a[2]0...0a[p] and b[1]0b[2]0...0b[q] are said to be equivalent if p=q and b[1],b[2],...,b[q] is a cyclic permutation of a[1],a[2],...a[p].
%C A132892 T(n,n) = A000108(n) (the Catalan numbers; see R. P. Stanley, Catalan addendum, problem starting "Equivalence classes of the equivalence relation ..."). T(m,m+1) = A007595(m+1); T(m,m+2) = A003441(m+1); T(m,m+3) = A003444(m+3); T(n+2,n) = A001453(n+1) (Catalan numbers - 1); T(m,1)=1; T(m,2)=m; T(m,3) = A080827(m) = A099392(m+1); T(m,4) = A004006(m).
%H A132892 Alois P. Heinz, <a href="/A132892/b132892.txt">antidiagonals n = 1..200, flattened</a>
%H A132892 Emeric Deutsch and Ira Gessel, <a href="https://www.jstor.org/stable/2589008">Equivalence Classes and Cyclic Arrangements:Problem 10525</a>, Amer. Math. Monthly, 105, No. 8, 1998, 774-775 (published solution by D. Beckwith).
%H A132892 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catadd.pdf">Catalan addendum</a>. See the interpretation (www, "Vertices of height n-1 of the tree T ...").
%F A132892 T(m,n) = Sum_{d | gcd(m,n+1)} phi(d)*(C((m+n+1)/d-1, (n+1)/d-1) - C(m/d-1, (n+1)/d-1))/(n+1). [corrected by _Jason Yuen_, Jan 28 2025]
%e A132892 T(2,4) = 3 because we have {2000, 0200, 0020, 0002}, {1100, 0110, 0011} and {1010, 0101, 1001}.
%e A132892 T(4,2) = 4 because we have {40, 04}, {31}, {13} and {22}.
%e A132892 The square array starts:
%e A132892 1....1.....1.....1......1.....1.....1...
%e A132892 1....2.....3.....3......4.....4.....5...
%e A132892 1....3.....5.....7.....10....12....15...
%e A132892 1....4.....9....14.....22....30....43...
%e A132892 1....5....13....25.....42....66....99...
%p A132892 with(numtheory): T:=proc(m,n) local r, div, N: r:=igcd(m,n+1): div:=divisors(r): N:=nops(div): (sum(phi(div[j])*(binomial((m+n+1)/div[j]-1,(n+1)/div[j]-1) -binomial(m/div[j]-1,(n+1)/div[j]-1)),j=1..N))/(n+1) end proc: for m to 12 do seq(T(m, n),n=1..12) end do; # yields the upper left 12 by 12 block of the infinite matrix T(m,n)
%p A132892 # second Maple program:
%p A132892 T:= proc(m, n) uses numtheory; (C-> add(phi(d)*(C((m+n+1)/d-1, (n+1)/d-1)
%p A132892 -C(m/d-1, (n+1)/d-1))/(n+1), d=divisors(igcd(m, n+1))))(binomial)
%p A132892 end:
%p A132892 seq(seq(T(1+d-n, n), n=1..d), d=1..14); # _Alois P. Heinz_, Jan 28 2025
%t A132892 T[m_, n_] := Module[{r, div, N}, r = GCD[m, n + 1]; div = Divisors[r]; N = Length[div]; (Sum[EulerPhi[div[[j]]]*(Binomial[(m + n + 1)/div[[j]] - 1, (n + 1)/div[[j]] - 1] - Binomial[m/div[[j]] - 1, (n + 1)/div[[j]] - 1]), {j, 1, N}])/(n + 1)];
%t A132892 Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Sep 01 2024, after Maple program *)
%Y A132892 Cf. A000108, A007595, A003441, A003444, A001453, A080827, A099392, A004006.
%K A132892 nonn,tabl
%O A132892 1,5
%A A132892 _Emeric Deutsch_ and _Ira M. Gessel_, Oct 02 2007