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

0,4

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

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.

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:

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)

Example

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;

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A321625 A132372 A103136 * A086960 A165675 A138771

Adjacent sequences: A155853 A155854 A155855 * A155857 A155858 A155859

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 29 2009

Status

approved