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 A262082 #23 Jun 03 2018 02:06:21
%S A262082 1,1,1,1,1,1,1,1,2,1,1,1,3,5,0,1,1,4,12,13,0,1,1,5,22,54,36,0,1,1,6,
%T A262082 35,139,262,104,0,1,1,7,51,284,953,1337,309,0,1,1,8,70,505,2509,6894,
%U A262082 7072,939,0,1,1,9,92,818,5455,23426,51796,38426,2905,0,1,1,10
%N A262082 Array of coefficients A(n,k) of the formal power series P(n,x) read by upwards antidiagonals, where P(n,x) = Sum_{k>=0} A(n,k)*x^k = 1 + x*P(n,x)^(1*n) + x^2*P(n,x)^(2*n) + x^3*P(n,x)^(3*n) for n >= 0.
%C A262082 The terms define the array A(n,k):
%C A262082 n\k: 0 1 2 3 4 5 6 7 8 9 10 ...
%C A262082 0: 1 1 1 1 0 0 0 0 0 0 0 ...
%C A262082 1: 1 1 2 5 13 36 104 309 939 2905 ...
%C A262082 2: 1 1 3 12 54 262 1337 7072 38426 ...
%C A262082 3: 1 1 4 22 139 953 6894 51796 400269 ...
%C A262082 4: 1 1 5 35 284 2509 23426 ...
%C A262082 5: 1 1 6 51 505 5455 62336 ...
%C A262082 6: 1 1 7 70 818 ...
%C A262082 7: 1 1 8 92 ...
%C A262082 8: 1 1 9 ...
%C A262082 9: 1 1 10 ...
%C A262082 10: 1 1 ...
%C A262082 11: 1 ...
%C A262082 etc.
%C A262082 For row 1 see A036765, for row 2 see A186241, and for row 3 see A200731.
%C A262082 Conjecture 1: The A(n,k), here n > 0, are the number of lattice paths, if
%C A262082 (a) length of path is k*n (for the k-th term of row n),
%C A262082 (b) allowed steps are (1,-1), (1,-1+n), (1,-1+2*n), and (1,-1+3*n),
%C A262082 (c) you start at (0,0), end at (k*n,0), and
%C A262082 (d) never cross the x-axis.
%C A262082 Conjecture 2: The coefficients B(m,n,k) of the P(n,x)^m (see the formula below), m > 0 and n > 0, are the number of lattice paths, if
%C A262082 (a) length of path is k*n+m-1 (k-th coefficient of P(n,x)^m),
%C A262082 (b) allowed steps are (1,-1), (1,-1+n), (1,-1+2*n), and (1,-1+3*n),
%C A262082 (c) you start at (0,m-1), end at (k*n+m-1,0), and
%C A262082 (d) never cross the x-axis.
%F A262082 A(n,k) = 1/(n*k+1) * Sum_{j=0..k} (-2)^j*binomial(n*k+1,j)* binomial(3*n*k+3-2*j,k-j) for n >= 0, and k >= 0. (conjectured)
%F A262082 A(n,0) = A(n,1) = 1, n >= 0;
%F A262082 A(n,2) = n+1, n >= 0;
%F A262082 A(n,3) = (3*n^2+5*n+2)/2, n >= 0;
%F A262082 A(n,4) = (8*n^3+18*n^2+13*n)/3, n >= 0;
%F A262082 A(n,5) = (125*n^4+350*n^3+355*n^2+34*n)/24, n >= 0.
%F A262082 The g.f. P(n,x) of row n of the array A(n,k) satisfy:
%F A262082 P(n,x) = P(n-1,x*P(n,x)), n > 0;
%F A262082 P(n,x) = P(n-2,x*P(n,x)^2), n > 1;
%F A262082 etc.
%F A262082 P(n,x) = P(0,x*P(n,x)^n), n >= 0.
%F A262082 The coefficients B(m,n,k) of the P(n,x)^m are:
%F A262082 B(m,n,k) = m/(n*k+m) * Sum_{j=0..k} (-2)^j*binomial(n*k+m,j)* binomial(3*n*k+3*m-2*j,k-j) for m > 0, n > 0, and k >= 0. (conjectured)
%F A262082 P(n,x) = exp(Sum_{k>=1} 1/(n*k)*(Sum_{j=0..k} (-2)^j*binomial(n*k,j)* binomial(3*n*k-2*j,k-j))) for n > 0 (conjectured); (see for n=1: A036765, for n=2: A186241, and for n=3: A200731).
%F A262082 P(n,x/(1+x+x^2+x^3)^n) = 1+x+x^2+x^3 for n >= 0. - _Werner Schulte_, Nov 20 2015
%e A262082 The terms of the array A(n,k) read by upwards antidiagonals define the triangle T(n,m) = A(n-m,m) for 0 <= m <= n, i.e.,
%e A262082 n\m 0 1 2 3 4 5 6 7 8 9 ...
%e A262082 0: 1
%e A262082 1: 1 1
%e A262082 2: 1 1 1
%e A262082 3: 1 1 2 1
%e A262082 4: 1 1 3 5 0
%e A262082 5: 1 1 4 12 13 0
%e A262082 6: 1 1 5 22 54 36 0
%e A262082 7: 1 1 6 35 139 262 104 0
%e A262082 8: 1 1 7 51 284 953 1337 309 0
%e A262082 9: 1 1 8 70 505 2509 6894 7072 939 0
%e A262082 etc. [reformatted by _Wolfdieter Lang_, Oct 15 2015]
%Y A262082 Cf. A036765, A186241, A200731, A261440.
%K A262082 nonn,tabl
%O A262082 0,9
%A A262082 _Werner Schulte_, Sep 10 2015