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A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1 <= k <= n.

%I A178300 #64 Apr 06 2025 19:02:16
%S A178300 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,
%T A178300 330,792,1716,1,9,45,165,495,1287,3003,6435,1,10,55,220,715,2002,5005,
%U A178300 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
%N A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1 <= k <= n.
%C A178300 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.
%C A178300 Also T(n,k) = count of degree k monomials in the Monomial symmetric polynomials m(mu,k) summed over all partitions mu of n.
%C A178300 T(n,k) is the number of ways to put n indistinguishable balls into k distinguishable boxes. - _Dennis P. Walsh_, Apr 11 2012
%C A178300 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
%C A178300 T(n,k) is the number of compositions (ordered partitions) of n+k into exactly k parts. - _Juergen Will_, Jan 23 2016
%C A178300 T(n,k) is the number of binary strings with exactly n zeros and k-1 ones. - _Dennis P. Walsh_, Apr 09 2016
%C A178300 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
%H A178300 P. A. MacMahon, <a href="http://www.jstor.org/stable/90632?seq=1#page_scan_tab_contents">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
%H A178300 Ch. Stover and E. W. Weisstein, <a href="https://mathworld.wolfram.com/Composition.html">Composition</a>. From MathWorld - A Wolfram Web Resource.
%H A178300 Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%F A178300 T(n,k) = A046899(n,k-1) = A038675(n,k)/A008292(n,k).
%F A178300 T(n,1) = 1.
%F A178300 T(n,2) = n+1.
%F A178300 T(n,3) = A000217(n+1).
%F A178300 T(n,4) = A000292(n+1).
%F A178300 T(n,5) = A000332(n+4).
%F A178300 T(n,n) = A001700(n-1) = A088218(n). - _Dennis P. Walsh_, Apr 10 2012
%e A178300 Triangle begins
%e A178300   1;
%e A178300   1,    3;
%e A178300   1,    4,   10;
%e A178300   1,    5,   15,   35;
%e A178300   1,    6,   21,   56,  126;
%e A178300   1,    7,   28,   84,  210,  462;
%e A178300   1,    8,   36,  120,  330,  792, 1716;
%e A178300 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
%e A178300 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
%p A178300 seq(seq(binomial(n+k-1,n),k=1..n),n=1..15); # _Dennis P. Walsh_, Apr 11 2012
%t A178300 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}]]];
%t A178300 Table[Tr[(m[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 11}, {k, l}](* _Wouter Meeussen_, Mar 11 2012 *)
%t A178300 Quiet[Needs["Combinatorica`"], All]; Grid[Table[Length[Combinatorica`Compositions[n, k]], {n, 10}, {k, n}]] (* _L. Edson Jeffery_, Jul 24 2014 *)
%t A178300 t[n_, k_] := Binomial[n + k - 1, n]; Table[ t[n, k], {n, 10}, {k, n}] // Flatten (* _Robert G. Wilson v_, Jul 24 2014 *)
%o A178300 (Magma) // As triangle
%o A178300 [[Binomial(n+k-1,n): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Jan 24 2016
%Y A178300 Cf. A000142, A000312, A007318, A001791 (row sums), A209664-A209673.
%Y A178300 Cf. A000217, A000292, A000332, A001700, A008292, A038675, A046899, A088218.
%Y A178300 Cf. A176992, A092392, A100100.
%K A178300 easy,nonn,tabl
%O A178300 1,3
%A A178300 _Alford Arnold_, May 24 2010