A059481 -id:A059481 - cp's OEIS frontend

A176992 Triangle T(n,m) = binomial(2n-k+1, n+1) read by rows, 0 <= k <= n.

1, 3, 1, 10, 4, 1, 35, 15, 5, 1, 126, 56, 21, 6, 1, 462, 210, 84, 28, 7, 1, 1716, 792, 330, 120, 36, 8, 1, 6435, 3003, 1287, 495, 165, 45, 9, 1, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1, 352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1

Offset: 0

Roger L. Bagula, Dec 08 2010

Row sums are A001791.
Obtained from A059481 by removal of the last two terms in each row, followed by row reversal.
Riordan array (c(x)/sqrt(1 - 4*x), x*c(x)) where c(x) is the g.f. of A000108. - Philippe Deléham, Jul 12 2015

			Triangle begins:
       1;
       3,      1;
      10,      4,     1;
      35,     15,     5,     1;
     126,     56,    21,     6,    1;
     462,    210,    84,    28,    7,     1;
    1716,    792,   330,   120,   36,     8,    1;
    6435,   3003,  1287,   495,   165,   45,    9,   1;
   24310,  11440,  5005,  2002,   715,  220,   55,  10,  1;
   92378,  43758, 19448,  8008,  3003, 1001,  286,  66, 11,  1;
  352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1;

Cf. A092392, A001791, A078812.
Cf. Similar triangle: A033184, A054445.
Cf. A178300 (reversal).

/* As triangle */ [[Binomial(2*n-k+1,n+1): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jul 12 2015

A176992 := proc(n,k) binomial(1+2*n-k,n+1) ; end proc: # R. J. Mathar, Dec 09 2010

p[t_, j_] = ((-1)^(j + 1)/2)*Sum[Binomial[k - j - 1, j + 1]*t^k, {k, 0, Infinity}];
Flatten[Table[CoefficientList[ExpandAll[p[t, j]], t], {j, 0, 10}]]

n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:
  3, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, ...
  ... - Philippe Deléham, Jul 12 2015

A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252, 462, 1, 7, 28, 84, 210, 462, 924, 1716, 1, 8, 36, 120, 330, 792, 1716, 3432, 6427, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12861, 24229, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24300, 48520, 91828

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 20 2012

For k <= 6, the triangle coincides with triangle A213745.

			Triangle begins
n/k |  0     1     2     3     4     5     6     7     8
----+---------------------------------------------------
  0 |  1
  1 |  1     1
  2 |  1     2     3
  3 |  1     3     6    10
  4 |  1     4    10    20    35
  5 |  1     5    15    35    70   126
  6 |  1     6    21    56   126   252   462
  7 |  1     7    28    84   210   462   924  1716
  8 |  1     8    36   120   330   792  1716  3432  6427

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A000580.

Table[Sum[(-1)^r*Binomial[n, r]*Binomial[n - 8*r + k - 1, n - 1], {r, 0, Floor[k/8]}], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, sum(r=0, floor(k/8), (-1)^r*binomial(n,r)*binomial(n-8*r + k-1,n-1))), ", "))) \\ G. C. Greubel, Nov 25 2017

T(n,k) = Sum_{r=0..floor(k/8)} (-1)^r*C(n,r)*C(n-8*r+k-1, n-1).

T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n > 1, T(n,3)=A000292(n) for n >= 3, T(n,4)=A000332(n) for n >= 7, T(n,5)=A000389(n) for n >= 9, T(n,6)=A000579(n) for n >= 11, T(n,7)=A000580(n) for n >= 13.

A130746 Triangle read by rows: T(n,m) = binomial(n+m,1+n), 1<=m<=n.

Original entry on oeis.org

1, 1, 4, 1, 5, 15, 1, 6, 21, 56, 1, 7, 28, 84, 210, 1, 8, 36, 120, 330, 792, 1, 9, 45, 165, 495, 1287, 3003, 1, 10, 55, 220, 715, 2002, 5005, 11440, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960

Offset: 1

Roger L. Bagula, Jul 12 2007

Row sums are A002054.

			1;
1, 4;
1, 5, 15;
1, 6, 21, 56;
1, 7, 28, 84, 210;
1, 8, 36, 120, 330, 792;
1, 9, 45, 165, 495, 1287, 3003;
1, 10, 55, 220, 715, 2002, 5005, 11440;

Cf. A178300, A046899, A059481, A002054, A000292.

Table[Table[Binomial[n + i, i + 1], {n, 1, i}], {i, 1, 10}] Flatten[%]

A174952 Triangle t(n,m)= binomial(n+m-1,n-1) + binomial(2n-m-1,n-1) -binomial(2n-1,n-1) read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -11, -15, -11, 1, 1, -51, -76, -76, -51, 1, 1, -204, -315, -350, -315, -204, 1, 1, -785, -1226, -1422, -1422, -1226, -785, 1, 1, -2995, -4683, -5523, -5775, -5523, -4683, -2995, 1, 1, -11431, -17830, -21142, -22528, -22528

Offset: 0

Roger L. Bagula, Apr 02 2010

Row sums are 1, 2, 3, 0, -35, -252, -1386, -6864, -32175, -145860, -646646,....

			1;
1, 1;
1, 1, 1;
1, -1, -1, 1;
1, -11, -15, -11, 1;
1, -51, -76, -76, -51, 1;
1 -204, -315, -350, -315, -204, 1;
1, -785, -1226, -1422, -1422, -1226, -785, 1;
1, -2995, -4683, -5523, -5775, -5523, -4683, -2995, 1;
1, -11431, -17830, -21142, -22528, -22528, -21142, -17830, -11431, 1;
1, -43748, -68013, -80718, -86658, -88374, -86658, -80718, -68013, -43748, 1;

Cf. A158498, A059481

A174952 := proc(n,m)
        binomial(n+m-1,n-1)+binomial(2*n-m-1,n-1) -binomial(2*n-1,n-1);
end proc: # R. J. Mathar, Jan 15 2013

t(n,m) = t(n,n-m).

Definition corrected and deobfuscated. R. J. Mathar, Jan 15 2013

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Original entry on oeis.org

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

The version for chains of divisors is A163767.
Diagonal n = k + 1 of A343658.
Choosing n divisors of n gives A343935.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343656, A343661.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]

a(n) = ((sigma(n - 1), n)) = binomial(sigma(n - 1) + n - 1, n) where sigma = A000005 and binomial = A007318.

A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

Original entry on oeis.org

1, 2, 1, 1, 3, 5, 5, 1, 4, 9, 16, 20, 1, 5, 14, 30, 55, 77, 1, 6, 20, 50, 105, 196, 294, 1, 7, 27, 77, 182, 378, 714, 1122, 1, 8, 35, 112, 294, 672, 1386, 2640, 4290, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445

Offset: 2

N. J. A. Sloane, Apr 24 2014

			Triangle begins:
1, 2, 1,
1, 3, 5, 5,
1, 4, 9, 16, 20,
1, 5, 14, 30, 55, 77,
1, 6, 20, 50, 105, 196, 294,
1, 7, 27, 77, 182, 378, 714, 1122,
1, 8, 35, 112, 294, 672, 1386, 2640, 4290,
1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445,
...

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, arXiv:1403.5827 [math.RT], 2014 and J. Int. Seq. 18 (2015) 15.10.6.

See A009766 for the case of type A.
See A059481 for the case of type B/C.
Diagonals give A029869, A051960, A029651, A051924. Row sums are also A051924.

f[t_, s_] := Binomial[t, s] (s + t)/t;
T[, 0] = 1; T[n, n_] := f[2 n - 2, n - 2]; T[n_, s_] := f[n + s - 2, s];
Table[T[n, s], {n, 2, 9}, {s, 0, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n,s) = [n+s-2,s] for 0 <= s < n, T(n,n) = [2n-2,n-2], where [t,s] stands for binomial(t,s)*(s+t)/t.

A316140 Denominator of the autosequence 2/((n+2)*(n+3)) difference table written by antidiagonals.

Original entry on oeis.org

3, 6, 6, 10, 15, 10, 15, 30, 30, 15, 21, 105, 70, 105, 21, 28, 84, 140, 140, 84, 28, 36, 126, 252, 315, 252, 126, 36, 45, 180, 420, 630, 630, 420, 180, 45, 55, 495, 660, 1155, 1386, 1155, 660, 495, 55, 66, 330

Offset: 0

Paul Curtz, Jun 25 2018

			Difference table:
   1/3,   1/6,    1/10,   1/15,  ...
  -1/6,  -1/15,  -1/30,  -2/105, ...
   1/10,  1/30,   1/70,   1/140, ...
  -1/15, -2/105, -1/140, -1/315, ... .
  ...
Table starts:
   3   6   10    15    21    28   ...
   6  15   30   105    84   126   ...
  10  30   70   140   252   420   ...
  15 105  140   315   630  1155   ...
  21  84  252   630  1386  2772   ...
  ...
As a triangle:
   3;
   6,  6;
  10, 15, 10;
  15, 30, 30, 15;
  ...

OEIS Wiki, Autosequence

Cf. A000217, A003506, A033876? (main diagonal), A059481, A109613.

tabl(nn) = {nn = 2*nn; m = matrix(nn, nn, n, k, if (n==1, 2/((k+1)*(k+2)))); for (n=2, nn, for (k=1, nn-n +1, m[n, k] = m[n-1, k+1] - m[n-1,k];);); nn = nn/2; matrix(nn, nn, n, k, denominator(m[n,k]));} \\ Michel Marcus, Jul 05 2018

cp's OEIS Frontend

A176992 Triangle T(n,m) = binomial(2n-k+1, n+1) read by rows, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A213808 Triangle of numbers C^(7)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 7 appearances allowed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A130746 Triangle read by rows: T(n,m) = binomial(n+m,1+n), 1<=m<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A174952 Triangle t(n,m)= binomial(n+m-1,n-1) + binomial(2*n-m-1,n-1) -binomial(2*n-1,n-1) read by rows, 0<=m<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A316140 Denominator of the autosequence 2/((n+2)*(n+3)) difference table written by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A174952 Triangle t(n,m)= binomial(n+m-1,n-1) + binomial(2n-m-1,n-1) -binomial(2n-1,n-1) read by rows, 0<=m<=n.