This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A186084 Triangle T(n,k) read by rows: number of 1-dimensional sand piles (see A186085) with n grains and base length k. 5
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS 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. Row sums are A186085. Column sums are the Motzkin numbers (A001006). First nonzero entry in row n appears in column A055086(n). From Joerg Arndt, Nov 06 2012: (Start) The transposed triangle (with zeros omitted) is A129181. 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). 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. (End) LINKS Alois P. Heinz, Rows n = 0..140, flattened Joerg Arndt, the first 36 rows as Pari script. FORMULA 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] G.f.: (formatting to make the structure apparent) A(x,y) = 1 / (1 - x^1*y / (1 - x^2*y / (1 - x^5*y^2 / (1 - x^3*y / (1 - x^4*y / (1 - x^9*y^2 / (1 - x^5*y / (1 - x^6*y / (1 - x^13*y^2 / (1 - x^7*y / (1 - x^8*y / (1 - x^17*y^2 / (1 -...))))))))))))) (continued fraction). [Paul D. Hanna] 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] EXAMPLE Triangle begins: 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; 0,0,0,0,1,6,5,1; 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; 0,0,0,0,0,1,7,19,35,28,9,1; MAPLE b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),       `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)): seq(T(n), n=0..20);  # Alois P. Heinz, Jul 24 2013 MATHEMATICA 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 *) PROG (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)} /* Paul D. Hanna */ CROSSREFS Cf. A186085 (sand piles with n grains, row sums), A001006 (Motzkin numbers, column sums), A055086. Cf. A186505 (antidiagonal sums). Sequence in context: A032239 A057094 A284938 * A047998 A017847 A127841 Adjacent sequences:  A186081 A186082 A186083 * A186085 A186086 A186087 KEYWORD nonn,tabl AUTHOR Joerg Arndt, Feb 13 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 18 17:56 EST 2017. Contains 294894 sequences.