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A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

%I A121757 #28 May 04 2017 05:54:23
%S A121757 1,1,2,1,4,6,1,6,18,24,1,8,36,96,120,1,10,60,240,600,720,1,12,90,480,
%T A121757 1800,4320,5040,1,14,126,840,4200,15120,35280,40320,1,16,168,1344,
%U A121757 8400,40320,141120,322560,362880,1,18,216,2016,15120,90720,423360,1451520
%N A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.
%C A121757 Row sums are 1,3,11,49,261,1631,... = A001339
%C A121757 a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns.  a(n,k) was discovered prior to array D. - _Roger Ford_, May 19 2016
%H A121757 Vincenzo Librandi, <a href="/A121757/b121757.txt">Rows n = 0..100, flattened</a>
%H A121757 J. Goldman, J. Haglund, <a href="http://dx.doi.org/10.1006/jcta.2000.3113">Generalized rook polynomials</a>, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
%H A121757 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A121757 a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - _R. J. Mathar_, Sep 02 2006
%F A121757 a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - _Franklin T. Adams-Watters_, Sep 20 2006
%e A121757 Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
%e A121757 From _Vincenzo Librandi_, Dec 16 2012: (Start)
%e A121757 Triangle begins:
%e A121757 1,
%e A121757 1, 2,
%e A121757 1, 4,  6,
%e A121757 1, 6,  18,  24,
%e A121757 1, 8,  36,  96,   120,
%e A121757 1, 10, 60,  240,  600,  720,
%e A121757 1, 12, 90,  480,  1800, 4320,  5040,
%e A121757 1, 14, 126, 840,  4200, 15120, 35280,  40320,
%e A121757 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
%e A121757 (End)
%t A121757 Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* _Harvey P. Dale_, Apr 25 2011 *)
%o A121757 (PARI) A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ _R. J. Mathar_, Sep 02 2006
%o A121757 (Haskell)
%o A121757 a121757 n k = a121757_tabl !! n !! k
%o A121757 a121757_row n = a121757_tabl !! n
%o A121757 a121757_tabl = iterate
%o A121757    (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
%o A121757 -- _Reinhard Zumkeller_, Mar 06 2014
%Y A121757 Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
%Y A121757 Cf. A008279.
%Y A121757 Cf. A008277, A132159 (mirrored).
%K A121757 nonn,easy,tabl
%O A121757 0,3
%A A121757 _Alford Arnold_, Aug 19 2006