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A178300
Triangle T(n, k) = binomial(n+k-1, n) read by rows, 1 <= k <= n.
7
1, 1, 3, 1, 4, 10, 1, 5, 15, 35, 1, 6, 21, 56, 126, 1, 7, 28, 84, 210, 462, 1, 8, 36, 120, 330, 792, 1716, 1, 9, 45, 165, 495, 1287, 3003, 6435, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078
OFFSET
1,3
COMMENTS
Obtained from A176992 by reversing entries in each row, from A092392 by removing the left column and reversing entries in each row, or from A100100 by removing the first two columns and reversing entries in each row.
Also T(n,k) = count of degree k monomials in the Monomial symmetric polynomials m(mu,k) summed over all partitions mu of n.
T(n,k) is the number of ways to put n indistinguishable balls into k distinguishable boxes. - Dennis P. Walsh, Apr 11 2012
T(n,k) is the number of compositions of n into k parts if zeros are allowed as parts. - L. Edson Jeffery, Jul 23 2014
T(n,k) is the number of compositions (ordered partitions) of n+k into exactly k parts. - Juergen Will, Jan 23 2016
T(n,k) is the number of binary strings with exactly n zeros and k-1 ones. - Dennis P. Walsh, Apr 09 2016
T(n,k) is the number of functions f:[k-1]->[n+1] that are nondecreasing. There is a unique correspondence between such a function and a binary string with exactly n zeros and k-1 ones. Given a string, let the corresponding function f be defined by f(i)=1 + (the number of zeros in the string that precede the i-th one in the string) for i=1,..,k-1. - Dennis P. Walsh, Apr 09 2016
LINKS
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Eric Weisstein's World of Mathematics, Composition.
FORMULA
T(n,k) = A046899(n,k-1) = A038675(n,k)/A008292(n,k).
T(n,1) = 1.
T(n,2) = n+1.
T(n,3) = A000217(n+1).
T(n,4) = A000292(n+1).
T(n,5) = A000332(n+4).
T(n,n) = A001700(n-1) = A088218(n). - Dennis P. Walsh, Apr 10 2012
From Natalia L. Skirrow, Oct 12 2025: (Start)
O.g.f.: y*f, where f(x,y) = ( 1 - (1-2*x)/sqrt(1-4*x*y) )/(2*(1-x-y)) is the solution to f*(1-(1-x-y)*f) = x/(1-4*x*y).
D-finite with k*T(n+1,k+1) = 2*(n+k)*T(n,k) + (k-n-1)*T(n+1,k).
O.g.f. of row n: y^n*C(2*n-1,n)*hypergeom([1,1-n],[1-2n],1/y).
O.g.f. of column n: x^k*C(2*k-1,k)*hypergeom([1,2*k],[k+1],x).
O.g.f. of diagonal n: x^(n+1)*y*hypergeom([n/2+1,(n+3)/2],[n+2],4*x*y). (End)
EXAMPLE
Triangle begins
1;
1, 3;
1, 4, 10;
1, 5, 15, 35;
1, 6, 21, 56, 126;
1, 7, 28, 84, 210, 462;
1, 8, 36, 120, 330, 792, 1716;
T(3,3)=10 since there are 10 ways to put 3 identical balls into 3 distinguishable boxes, namely, (OOO)()(), ()(OOO)(), ()()(OOO), (OO)(O)(), (OO)()(O), (O)(OO)(), ()(OO)(O), (O)()(OO), ()(O)(OO), and (O)(O)(O). - Dennis P. Walsh, Apr 11 2012
For example, T(3,3)=10 since there are ten functions f:[2]->[4] that are nondecreasing, namely, <f(1),f(2)> = <1,1> or <1,2> or <1,3> or <1,4> or <2,2> or <2,3> or <2,4> or <3,3> or <3,4> or <4,4>. - Dennis P. Walsh, Apr 09 2016
MAPLE
seq(seq(binomial(n+k-1, n), k=1..n), n=1..15); # Dennis P. Walsh, Apr 11 2012
gf := y*(1 - (1 - 2*x)/sqrt(1 - 4*x*y) )/(2*(1 - x - y)):
Max := 10: ser := simplify(series(gf, x, Max + 1)):
seq(print(seq(coeff(coeff(ser, x, n), y, k), k = 1..n)), n = 1..Max); # Peter Luschny, Oct 13 2025
MATHEMATICA
m[par_?PartitionQ, v_] := Block[{le = Length[par], it }, If[le > v, Return[0]]; it = Permutations[PadRight[par, v]]; Tr[ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ it, {1}]]];
Table[Tr[(m[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 11}, {k, l}](* Wouter Meeussen, Mar 11 2012 *)
Quiet[Needs["Combinatorica`"], All]; Grid[Table[Length[Combinatorica`Compositions[n, k]], {n, 10}, {k, n}]] (* L. Edson Jeffery, Jul 24 2014 *)
t[n_, k_] := Binomial[n + k - 1, n]; Table[ t[n, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Jul 24 2014 *)
PROG
(Magma) // As triangle
[[Binomial(n+k-1, n): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 24 2016
KEYWORD
easy,nonn,tabl
AUTHOR
Alford Arnold, May 24 2010
STATUS
approved