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

%I A155856 #15 Aug 16 2025 07:04:25
%S A155856 1,1,1,2,3,1,6,10,6,1,24,42,30,10,1,120,216,168,70,15,1,720,1320,1080,
%T A155856 504,140,21,1,5040,9360,7920,3960,1260,252,28,1,40320,75600,65520,
%U A155856 34320,11880,2772,420,36,1,362880,685440,604800,327600,120120,30888,5544,660,45,1
%N A155856 Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.
%C A155856 Row sums of B^{-1}*A155856*B^{-1} are A000166 with B=A007318.
%C A155856 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
%H A155856 G. C. Greubel, <a href="/A155856/b155856.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A155856 Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.
%F A155856 T(n,k) = binomial(2*n-k, k)*(n-k)!.
%F A155856 Sum_{k=0..n} T(n, k) = A155857(n)
%F A155856 Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
%F A155856 G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
%F A155856 From _G. C. Greubel_, Jun 04 2021: (Start)
%F A155856   T(n, 0) = A000142(n).        T(n+1, n) = A000217(n+1).
%F A155856 T(n+1, 1) = A007680(n).        T(n+2, n) = A034827(n+4).
%F A155856 T(n+2, 2) = A175925(n).        T(n+3, n) = A253946(n).
%F A155856 T(2*n, n) = A064352(n)         T(n+4, n) = 4!*A000581(n).
%F A155856 T(n+1, n) = A000217(n+1).      T(n+5, n) = 5!*A001287(n).  (End)
%e A155856 Triangle begins:
%e A155856      1;
%e A155856      1,    1;
%e A155856      2,    3,    1;
%e A155856      6,   10,    6,    1;
%e A155856     24,   42,   30,   10,    1;
%e A155856    120,  216,  168,   70,   15,   1;
%e A155856    720, 1320, 1080,  504,  140,  21,  1;
%e A155856   5040, 9360, 7920, 3960, 1260, 252, 28, 1;
%t A155856 Table[Binomial[2n-k,k](n-k)!,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Mar 24 2017 *)
%o A155856 (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
%Y A155856 Cf. A155857 (row sums), A155858 (diagonal sums).
%K A155856 easy,nonn,tabl
%O A155856 0,4
%A A155856 _Paul Barry_, Jan 29 2009