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A193739 Mirror of the triangle A193738.

%I A193739 #25 Aug 17 2021 02:47:32
%S A193739 1,1,1,2,2,1,3,3,2,1,4,4,3,2,1,5,5,4,3,2,1,6,6,5,4,3,2,1,7,7,6,5,4,3,
%T A193739 2,1,8,8,7,6,5,4,3,2,1,9,9,8,7,6,5,4,3,2,1,10,10,9,8,7,6,5,4,3,2,1,11,
%U A193739 11,10,9,8,7,6,5,4,3,2,1,12,12,11,10,9,8,7,6,5,4,3,2,1,13,13,12
%N A193739 Mirror of the triangle A193738.
%C A193739 This sequence is obtained by reversing the rows of the triangle A193738.
%C A193739 Except for the first term, this sequence gives the integers occurring in the song "One man went to mow".
%H A193739 Reinhard Zumkeller, <a href="/A193739/b193739.txt">Rows n = 0..100 of triangle, flattened</a>
%H A193739 BBC, <a href="https://www.bbc.co.uk/teach/school-radio/nursery-rhymes-one-man-went-to-mow/zjgnmfr">One man went to mow.</a>
%H A193739 <a href="/index/So#songs">Index entries for sequences related to songs</a>
%F A193739 Write w(n,k) for the triangle at A193738.  The current triangle is then given by w(n,n-k).
%e A193739 First six rows:
%e A193739 1
%e A193739 1....1
%e A193739 2....2....1
%e A193739 3....3....2....1
%e A193739 4....4....3....2...1
%e A193739 5....5....4....3...2...1
%t A193739 z = 12;
%t A193739 p[0, x_] := 1
%t A193739 p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0
%t A193739 q[n_, x_] := p[n, x]
%t A193739 t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
%t A193739 t[n_, n_] := p[n, x] /. x -> 0;
%t A193739 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193739 g[n_] := CoefficientList[w[n, x], {x}]
%t A193739 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193739 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193738 *)
%t A193739 TableForm[Table[g[n], {n, -1, z}]]
%t A193739 Flatten[Table[g[n], {n, -1, z}]]   (* A193739 *)
%o A193739 (Haskell)
%o A193739 a193738 n k = a193738_tabl !! n !! k
%o A193739 a193738_row n = a193738_tabl !! n
%o A193739 a193738_tabl = map reverse a193739_tabl
%o A193739 -- _Reinhard Zumkeller_, May 11 2013
%Y A193739 Cf. A193738.
%K A193739 nonn,tabl
%O A193739 0,4
%A A193739 _Clark Kimberling_, Aug 04 2011