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A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.

%I A186084 #46 Mar 11 2022 07:45:09
%S A186084 1,0,1,0,0,1,0,0,1,1,0,0,0,2,1,0,0,0,1,3,1,0,0,0,0,3,4,1,0,0,0,0,1,6,
%T A186084 5,1,0,0,0,0,1,4,10,6,1,0,0,0,0,0,3,10,15,7,1,0,0,0,0,0,2,8,20,21,8,1,
%U A186084 0,0,0,0,0,1,7,19,35,28,9,1,0,0,0,0,0,0,5,18,40,56,36,10,1,0,0,0,0,0,0,3,16,41,76,84,45,11,1,0,0,0,0,0,0,1,12,41,86,133,120,55,12,1
%N A186084 Triangle T(n,k) read by rows: number of 1-dimensional sandpiles (see A186085) with n grains and base length k.
%C A186084 Compositions of n into k nonzero parts such that the first and last parts are 1 and the absolute difference between consecutive parts is <=1.
%C A186084 Row sums are A186085.
%C A186084 Column sums are the Motzkin numbers (A001006).
%C A186084 First nonzero entry in row n appears in column A055086(n).
%C A186084 From _Joerg Arndt_, Nov 06 2012: (Start)
%C A186084 The transposed triangle (with zeros omitted) is A129181.
%C A186084 For large k, the columns end in reverse([1, 1, 3, 5, 9, 14, 24, 35, ...]) for k even (cf. A053993) and reverse([1, 2, 3, 6, 10, 16, 26, 40, 60, 90, ...]) for k odd (cf. A201077).
%C A186084 The diagonals below the main diagonal are (apart from leading zeros), n, n*(n+1)/2, n*(n+1)*(n+2)/6, and the e-th diagonal appears to have a g.f. of the form f(x)/(1-x)^e.
%C A186084 (End)
%H A186084 Alois P. Heinz, <a href="/A186084/b186084.txt">Rows n = 0..140, flattened</a>
%H A186084 Joerg Arndt, <a href="/A186084/a186084.gp.txt">the first 36 rows</a> as Pari script.
%F A186084 G.f. A(x,y) satisfies:  A(x,y) = 1/(1 - x*y - x^3*y^2*A(x, x*y) ). [_Paul D. Hanna_, Feb 22 2011]
%F A186084 G.f.: (formatting to make the structure apparent)
%F A186084 A(x,y) = 1 /
%F A186084 (1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 /
%F A186084 (1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 /
%F A186084 (1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 /
%F A186084 (1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...)))))))))))))
%F A186084 (continued fraction). [_Paul D. Hanna_]
%F A186084 G.f.: A(x,y) = 1/(1-x*y - x^3*y^2/(1-x^2*y - x^5*y^2/(1-x^3*y - x^7*y^2/(1 -...)))) (continued fraction). [_Paul D. Hanna_]
%e A186084 Triangle begins:
%e A186084 1;
%e A186084 0,1;
%e A186084 0,0,1;
%e A186084 0,0,1,1;
%e A186084 0,0,0,2,1;
%e A186084 0,0,0,1,3,1;
%e A186084 0,0,0,0,3,4,1;
%e A186084 0,0,0,0,1,6,5,1;
%e A186084 0,0,0,0,1,4,10,6,1;
%e A186084 0,0,0,0,0,3,10,15,7,1;
%e A186084 0,0,0,0,0,2,8,20,21,8,1;
%e A186084 0,0,0,0,0,1,7,19,35,28,9,1;
%p A186084 b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
%p A186084       `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
%p A186084     end:
%p A186084 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
%p A186084 seq(T(n), n=0..20);  # _Alois P. Heinz_, Jul 24 2013
%t A186084 b[n_, i_] := b[n, i] = If[n == 0, If[i == 1, 1, 0], If[n<0 || i<1, 0, Expand[ x*Sum[b[n-i, i+j], {j, -1, 1}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%o A186084 (PARI) {T(n,k)=local(A=1+x*y);for(i=1,n,A=1/(1-x*y-x^3*y^2*subst(A,y,x*y+x*O(x^n))));polcoeff(polcoeff(A,n,x),k,y)}
%o A186084 /* _Paul D. Hanna_ */
%Y A186084 Cf. A186085 (sandpiles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086.
%Y A186084 Cf. A186505 (antidiagonal sums).
%K A186084 nonn,tabl
%O A186084 0,14
%A A186084 _Joerg Arndt_, Feb 13 2011