A155856 Triangle T(n,k) = binomial(2*n-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

Author

Paul Barry, Jan 29 2009

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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;
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Sage

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

Formula

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)