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A193091 Augmentation of the triangular array A158405. See Comments.

%I A193091 #25 Jan 05 2025 19:51:39
%S A193091 1,1,3,1,6,14,1,9,37,79,1,12,69,242,494,1,15,110,516,1658,3294,1,18,
%T A193091 160,928,3870,11764,22952,1,21,219,1505,7589,29307,85741,165127,1,24,
%U A193091 287,2274,13355,61332,224357,638250,1217270,1,27,364,3262,21789,115003
%N A193091 Augmentation of the triangular array A158405. See Comments.
%C A193091 Suppose that P is an infinite triangular array of numbers:
%C A193091 p(0,0)
%C A193091 p(1,0)...p(1,1)
%C A193091 p(2,0)...p(2,1)...p(2,2)
%C A193091 p(3,0)...p(3,1)...p(3,2)...p(3,3)...
%C A193091 ...
%C A193091 Let w(0,0)=1, w(1,0)=p(1,0), w(1,1)=p(1,1), and define
%C A193091     W(n)=(w(n,0), w(n,1), w(n,2),...w(n,n-1), w(n,n)) recursively by W(n)=W(n-1)*PP(n), where PP(n) is the n X (n+1) matrix given by
%C A193091 ...
%C A193091 row 0 ... p(n,0) ... p(n,1) ...... p(n,n-1) ... p(n,n)
%C A193091 row 1 ... 0 ..... p(n-1,0) ..... p(n-1,n-2) .. p(n-1,n-1)
%C A193091 row 2 ... 0 ..... 0 ............ p(n-2,n-3) .. p(n-2,n-2)
%C A193091 ...
%C A193091 row n-1 . 0 ..... 0 ............. p(2,1) ..... p(2,2)
%C A193091 row n ... 0 ..... 0 ............. p(1,0) ..... p(1,1)
%C A193091 ...
%C A193091 The augmentation of P is here introduced as the triangular array whose n-th row is W(n), for n>=0. The array P may be represented as a sequence of polynomials; viz., row n is then the vector of coefficients:  p(n,0), p(n,1),...,p(n,n), from p(n,0)*x^n+p(n,1)*x^(n-1)+...+p(n,n). For example, (C(n,k)) is represented by ((x+1)^n); using this choice of P (that is, Pascal's triangle), the augmentation of P is calculated one row at a time, either by the above matrix products or by polynomial substitutions in the following manner:
%C A193091 ...
%C A193091 row 0 of W:  1, by decree
%C A193091 row 1 of W:  1 augments to 1,1
%C A193091 ...polynomial version:  1 -> x+1
%C A193091 row 2 of W:  1,1 augments to 1,3,2
%C A193091 ...polynomial version:  x+1 -> (x^2+2x+1)+(x+1)=x^2+3x+2
%C A193091 row 3 to W:  1,3,2 augments to 1,6,11,6
%C A193091 ...polynomial version:
%C A193091     x^2+3x+2 -> (x+1)^3+3(x+1)^2+2(x+1)=(x+1)(x+2)(x+3)
%C A193091 ...
%C A193091 Examples of augmented triangular arrays:
%C A193091 (p(n,k)=1) augments to A009766, Catalan triangle.
%C A193091 Catalan triangle augments to A193560.
%C A193091 Pascal triangle augments to A094638, Stirling triangle.
%C A193091 A002260=((k+1)) augments to A023531.
%C A193091 A154325 augments to A033878.
%C A193091 A158405 augments to A193091.
%C A193091 ((k!)) augments to A193092.
%C A193091 A094727 augments to A193093.
%C A193091 A130296 augments to A193094.
%C A193091 A004736 augments to A193561.
%C A193091 ...
%C A193091 Regarding the specific augmentation W=A193091: w(n,n)=A003169.
%C A193091 From _Peter Bala_, Aug 02 2012: (Start)
%C A193091 This is the table of g(n,k) in the notation of Carlitz (p. 124). The triangle enumerates two-line arrays of positive integers
%C A193091 ............a_1 a_2 ... a_n..........
%C A193091 ............b_1 b_2 ... b_n..........
%C A193091 such that
%C A193091 1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
%C A193091 2) max(a_i, b_i) <= i for 1 <= i <= n
%C A193091 3) max(a_n, b_n) = k.
%C A193091 See A071948 and A211788 for other two-line array enumerations.
%C A193091 (End)
%H A193091 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
%F A193091 From _Peter Bala_, Aug 02 2012: (Start)
%F A193091 T(n,k) = (n-k+1)/n*Sum_{i=0..k} C(n+1,n-k+i+1)*C(2*n+i+1,i) for 0 <= k <= n.
%F A193091 Recurrence equation: T(n,k) = Sum_{i=0..k} (2*k-2*i+1)*T(n-1,i).
%F A193091 (End)
%e A193091 The triangle P, at A158405, is given by rows
%e A193091   1
%e A193091   1...3
%e A193091   1...3...5
%e A193091   1...3...5...7
%e A193091   1...3...5...7...9...
%e A193091 The augmentation of P is the array W starts with w(0,0)=1, by definition of W.
%e A193091 Successive polynomials (rows of W) arise from P as shown here:
%e A193091   ...
%e A193091   1->x+3, so that W has (row 1)=(1,3);
%e A193091   ...
%e A193091   x+3->(x^2+3x+5)+3*(x+3), so that W has (row 2)=(1,6,14);
%e A193091   ...
%e A193091   x^2+6x+14->(x^3+3x^2+5x+7)+6(x^2+3x+5)+14(x+3), so that (row 3)=(1,9,37,79).
%e A193091   ...
%e A193091 First 7 rows of W:
%e A193091   1
%e A193091   1    3
%e A193091   1    6    14
%e A193091   1    9    37    79
%e A193091   1   12    69   242    494
%e A193091   1   15   110   516   1658    3294
%e A193091   1   18   160   928   3870   11764   22952
%t A193091 p[n_, k_] := 2 k + 1
%t A193091 Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A158405 *)
%t A193091 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193091 TableForm[m[4]]
%t A193091 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193091 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193091 v[n_] := v[n - 1].m[n]
%t A193091 TableForm[Table[v[n], {n, 0, 6}]] (* A193091 *)
%t A193091 Flatten[Table[v[n], {n, 0, 9}]]
%Y A193091 Cf. A003169, A193092, A193093, A193094.
%Y A193091 Cf. A071948, A211788.
%K A193091 nonn,tabl,easy
%O A193091 0,3
%A A193091 _Clark Kimberling_, Jul 30 2011