A038675 -id:A038675 - cp's OEIS frontend

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

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

Alford Arnold, May 24 2010

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

			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

P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Ch. Stover and E. W. Weisstein, Composition. From MathWorld - A Wolfram Web Resource.
Wikipedia, Symmetric Polynomials

Cf. A000142, A000312, A007318, A001791 (row sums), A209664-A209673.
Cf. A000217, A000292, A000332, A001700, A008292, A038675, A046899, A088218.
Cf. A176992, A092392, A100100.

// As triangle
[[Binomial(n+k-1,n): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 24 2016

seq(seq(binomial(n+k-1,n),k=1..n),n=1..15); # Dennis P. Walsh, Apr 11 2012

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 *)

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

A178302 Multiply the irregular Array A125108 by A178300;compute a(n)the vertical sums.

Original entry on oeis.org

1, 4, 19, 104, 601, 3622

Offset: 1

Alford Arnold, May 30 2010

The row sums resulting from the defined multiplication yields A038675.
A178301 is a triangular sub-array of A178300 times A125108
since A007318 is a sub-array of A125108.

			A125108(7) = 2 and appears on row five of A125108 so
A178300(5) times A125108(7) is 4*2 =8.
As a cross-check, note that A178301 = 1,4,19,96,...
and with the additional 8 in column 4 we have a(n) = 1,4,19,104,...

Cf. A038675 A125108 A178300 A178301

cp's OEIS Frontend

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

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