A155857 -id:A155857 - cp's OEIS frontend

A155856 Triangle T(n,k) = binomial(2n-k, k)(n-k)!, read by rows.

1, 1, 1, 2, 3, 1, 6, 10, 6, 1, 24, 42, 30, 10, 1, 120, 216, 168, 70, 15, 1, 720, 1320, 1080, 504, 140, 21, 1, 5040, 9360, 7920, 3960, 1260, 252, 28, 1, 40320, 75600, 65520, 34320, 11880, 2772, 420, 36, 1, 362880, 685440, 604800, 327600, 120120, 30888, 5544, 660, 45, 1

Offset: 0

Paul Barry, Jan 29 2009

Row sums of B^{-1}*A155856*B^{-1} are A000166 with B=A007318.

Downward diagonals T(n+j, n) = j!*binomial(n+j, n) = j!*seq(j), where seq(j) are sequences A010965, A010967, ..., A011101, A017714, A017716, ..., A017764, for 6 <= j <= 50, respectively. - G. C. Greubel, Jun 04 2021

			Triangle begins:
     1;
     1,    1;
     2,    3,    1;
     6,   10,    6,    1;
    24,   42,   30,   10,    1;
   120,  216,  168,   70,   15,   1;
   720, 1320, 1080,  504,  140,  21,  1;
  5040, 9360, 7920, 3960, 1260, 252, 28, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Cf. A155857 (row sums), A155858 (diagonal sums).

Table[Binomial[2n-k,k](n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 24 2017 *)

flatten([[factorial(n-k)*binomial(2*n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n,k) = binomial(2*n-k, k)*(n-k)!.
Sum_{k=0..n} T(n, k) = A155857(n)
Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
From G. C. Greubel, Jun 04 2021: (Start)
  T(n, 0) = A000142(n).        T(n+1, n) = A000217(n+1).
T(n+1, 1) = A007680(n).        T(n+2, n) = A034827(n+4).
T(n+2, 2) = A175925(n).        T(n+3, n) = A253946(n).
T(2*n, n) = A064352(n)         T(n+4, n) = 4!*A000581(n).
T(n+1, n) = A000217(n+1).      T(n+5, n) = 5!*A001287(n).  (End)

A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

Original entry on oeis.org

1, 3, 9, 32, 139, 729, 4515, 32336, 263205, 2401183, 24275037, 269426592, 3257394143, 42615550453, 599875100487, 9040742057760, 145251748024649, 2478320458476795, 44755020000606961, 852823700470009056, 17101229029400788083, 359978633317886558801, 7936631162022905081707

Offset: 2

Tamas Dekany, May 26 2016

			The initial term is the diagram of the four element diamond shape lattice.

Vaclav Kotesovec, Table of n, a(n) for n = 2..400
P. Bala, Notes on A273596
Gábor Czédli, Tamás Dékány, Gergő Gyenizse, and Júlia Kulin, The number of slim rectangular lattices, Algebra Universalis, 2016, Volume 75, Issue 1, pp 33-50.

Cf. A273988, A000522, A143409, A000179, A000271, A000904, A127548.

Partial sums of A155857.

A273596 := proc (n) option remember; `if`(n = 2, 1, `if`(n = 3, 3, (n-2)*procname(n-1) + procname(n-2) + 2)) end: seq(A273596(n), n = 2..20); # Peter Bala, Jan 08 2017

x = 15;
SRectD = Table[0, {x}];
For[n = 2, n < x, n++,
For[a = 1, a < n, a++,
   For[b = 1, b <= n - a, b++,
    SRectD[[n]] +=
      Binomial[n - a - 1, b - 1]*
       Binomial[n - b - 1, a - 1]*(n - a - b)!;
    ]
   ]
  Print[n, " ", SRectD[[n]]]
]
(* Alternatively: *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];
Table[Sum[T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

a(n)= sum(rps=1, n, sum(r=1, n, s = rps-r; binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!)); \\ Michel Marcus, Jun 12 2016

a(n) = Sum_{1<=r,s; r+s<=n} binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!.
a(n) ~ exp(2) * n! / n^2. - Vaclav Kotesovec, Jun 29 2016
a(n) = Sum_{k=0..n} hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
From Peter Bala, Jan 08 2018: (Start)
a(n) = Sum_{k = 0..n-2} k!*binomial(n+k-1, 2*k+1).
a(n) = (n - 2)*a(n-1) + a(n-2) + 2, with a(2) = 1, a(3) = 3.
a(n+2) = 1/n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)* A000522(n)^2.
Row sums of array A143409 read as a triangle.
O.g.f.: Sum_{n >= 0} n!*x^(n+2)/(1 - x)^(2*n+2). Cf. A000179, A000271, A000904 and A127548.
O.g.f. with offset 0: 1/(1 - x) o 1/(1 - x) = 1 + 3*x + 9*x^2 + 32*x^3 + ..., where o denotes the white diamond multiplication of power series. See the Bala link for details. (End)

A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 24, 1, 15, 70, 168, 216, 120, 1, 21, 140, 504, 1080, 1320, 720, 1, 28, 252, 1260, 3960, 7920, 9360, 5040, 1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320, 1, 45, 660, 5544, 30888, 120120, 327600, 604800, 685440, 362880

Offset: 0

Paul Barry, Feb 08 2009

			Triangle begins
  1;
  1,  1;
  1,  3,   2;
  1,  6,  10,    6;
  1, 10,  30,   42,    24;
  1, 15,  70,  168,   216,   120;
  1, 21, 140,  504,  1080,  1320,   720;
  1, 28, 252, 1260,  3960,  7920,  9360,  5040;
  1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320;

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A084261 (diagonal sums), A155856 (row reversal), A155857 (row sums)

Flatten[Table[Binomial[n+k,2k]k!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 17 2015 *)

flatten([[factorial(k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

G.f.: 1/(1 -x -x*y/(1 -x -x*y/(1 -x -2*x*y/(1 -x -2*x*y/(1 -x -3*x*y/(1 -x -3*x*y/(1 - ... (continued fraction).
T(n, k) = binomial(n+k, 2*k)*k!
T(n, k) = A155856(n, n-k).
Sum_{k=0..n} T(n, k) = A155857(n).
sum_{k=0..floor(n/2)} T(n, k) = A084261(n).

cp's OEIS Frontend

A155856 Triangle T(n,k) = binomial(2n-k, k)(n-k)!, read by rows.

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A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

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A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.

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cp's OEIS Frontend

A155856 Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A156367 Triangle T(n, k) = binomial(n+k, 2*k)*k!, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A155856 Triangle T(n,k) = binomial(2n-k, k)(n-k)!, read by rows.

A156367 Triangle T(n, k) = binomial(n+k, 2k)k!, read by rows.