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A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.

%I A093375 #88 Aug 27 2024 02:54:14
%S A093375 1,2,1,3,4,1,4,9,6,1,5,16,18,8,1,6,25,40,30,10,1,7,36,75,80,45,12,1,8,
%T A093375 49,126,175,140,63,14,1,9,64,196,336,350,224,84,16,1,10,81,288,588,
%U A093375 756,630,336,108,18,1,11,100,405,960,1470,1512,1050,480,135,20,1,12
%N A093375 Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
%C A093375 Number of n-long m-ary words avoiding the pattern 1-1'2'.
%C A093375 T(n,n+1) = Sum_{i=1..n} T(n,i).
%C A093375 Exponential Riordan array [(1+x)e^x, x] as a number triangle. - _Paul Barry_, Feb 17 2009
%C A093375 From _Peter Bala_, Jul 22 2014: (Start)
%C A093375 Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C A093375 /I_k 0\
%C A093375 \ 0  M/
%C A093375 having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A059298. (End)
%H A093375 Muniru A Asiru, <a href="/A093375/b093375.txt">Antidiagonals, n=1..100 flattened</a>
%H A093375 Yasemin Alp and E. Gokcen Kocer, <a href="https://doi.org/10.1007/s00025-024-02193-5">Exponential Almost-Riordan Arrays</a>, Results Math 79, 173 (2024). See page 7.
%H A093375 Sergey Kitaev and Toufik Mansour, <a href="https://arxiv.org/abs/math/0210023">Partially ordered generalized patterns and k-ary words</a>, arXiv:math/0210023 [math.CO], 2002.
%H A093375 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sheffer_sequence">Sheffer sequence</a>.
%F A093375 Triangle = P*M, the binomial transform of the infinite bidiagonal matrix M with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal, and zeros elsewhere. P = Pascal's triangle as an infinite lower triangular matrix. - _Gary W. Adamson_, Nov 05 2006
%F A093375 From _Peter Bala_, Sep 20 2012: (Start)
%F A093375 E.g.f. for triangle: (1 + z)*exp((1 + x)*z) = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2/2! + ....
%F A093375 O.g.f. for triangle: (1 - x*z)/(1 - z - x*z)^2 = 1 + (2 + x)*z + (3 + 4*x + x^2)*z^2 + ....
%F A093375 The n-th row polynomial R(n,x) of the triangle equals (1+x)^n + n*(1+x)^(n-1) for n >= 0 and satisfies d/dx(R(n,x)) = n*R(n-1,x), as well as R(n,x+y) = Sum_{k = 0..n} binomial(n,k)*R(k,x)*y^(n-k). The row polynomials are a Sheffer sequence of Appell type.
%F A093375 Matrix inverse of the triangle is a signed version of A073107. (End)
%F A093375 From _Tom Copeland_, Oct 20 2015: (Start)
%F A093375 With offset 0 and D = d/dx, the raising operator for the signed row polynomials P(n,x) is RP = x - d{log[e^D/(1-D)]}/dD = x - 1 - 1/(1-D) =  x - 2 - D - D^2 + ..., i.e., RP P(n,x) = P(n+1,x).
%F A093375 The e.g.f. for the signed array is (1-t) * e^(-t) * e^(x*t).
%F A093375 From the Appell formalism, the row polynomials PI(n,x) of A073107 are the umbral inverse of this entry's row polynomials; that is, P(n,PI(.,x)) = x^n = PI(n,P(.,x)) under umbral composition. (End)
%F A093375 From _Petros Hadjicostas_, Nov 01 2019: (Start)
%F A093375 As a triangle, we let S(n,k) = T(n-k+1, k+1) = (n-k+1)*binomial(n, k) for n >= 0 and 0 <= k <= n. See the example below.
%F A093375 As stated above by _Peter Bala_, Sum_{n,k >= 0} S(n,k)*z^n*x^k = (1 - x*z)/(1 - z -x*z)^2.
%F A093375 Also, Sum_{n, k >= 0} S(n,k)*z^n*x^k/n! = (1+z)*exp((1+x)*z).
%F A093375 As he also states, the n-th row polynomial is R(n,x) = Sum_{k = 0..n} S(n, k)*x^k = (1 + x)^n + n*(1 + x)^(n-1).
%F A093375 If we define the signed triangle S*(n,k) = (-1)^(n+k) * S(n,k) = (-1)^(n+k) * T(n-k+1, k+1), as _Tom Copeland_ states, Sum_{n,k >= 0} S^*(n,k)*t^n*x^k/n! = (1-t)*exp((1-x)*(-t)) = (1-t) * e^(-t) * e^(x*t).
%F A093375 Apparently, S*(n,k) = A103283(n,k).
%F A093375 As he says above, the signed n-th row polynomial is P(n,x) =  (-1)^n*R(n,-x) = (x - 1)^n - n*(x - 1)^(n-1).
%F A093375 According to _Gary W. Adamson_, P(n,x) is "the monic characteristic polynomial of the n X n matrix with 2's on the diagonal and 1's elsewhere." (End)
%e A093375 Array T(m,n) (with rows m >= 1 and columns n >= 1) begins as follows:
%e A093375    1   1   1   1   1   1 ...
%e A093375    2   4   6   8  10  12 ...
%e A093375    3   9  18  30  45  63 ...
%e A093375    4  16  40  80 140 224 ...
%e A093375    5  25  75 175 350 630 ...
%e A093375    ...
%e A093375 Triangle S(n,k) = T(n-k+1, k+1) begins
%e A093375 .n\k.|....0....1....2....3....4....5....6
%e A093375 = = = = = = = = = = = = = = = = = = = = =
%e A093375 ..0..|....1
%e A093375 ..1..|....2....1
%e A093375 ..2..|....3....4....1
%e A093375 ..3..|....4....9....6....1
%e A093375 ..4..|....5...16...18....8....1
%e A093375 ..5..|....6...25...40...30...10....1
%e A093375 ..6..|....7...36...75...80...45...12....1
%e A093375 ...
%t A093375 nmax = 10;
%t A093375 T = Transpose[CoefficientList[# + O[z]^(nmax+1), z]& /@ CoefficientList[(1 - x z)/(1 - z - x z)^2 + O[x]^(nmax+1), x]];
%t A093375 row[n_] := T[[n+1, 1 ;; n+1]];
%t A093375 Table[row[n], {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%o A093375 (GAP) nmax:=14;; T:=List([1..nmax],n->List([1..nmax],k->k*Binomial(n+k-2,n-1)));;
%o A093375 b:=List([2..nmax],n->OrderedPartitions(n,2));;
%o A093375 a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][1]][b[i][j][2]]))); # _Muniru A Asiru_, Aug 07 2018
%o A093375 (Sage) # uses[riordan_array from A256893]
%o A093375 riordan_array((1+x)*exp(x), x, 8, exp=true) # _Peter Luschny_, Nov 02 2019
%Y A093375 Rows include A045943. Columns include A002411, A027810.
%Y A093375 Main diagonal is A037965. Subdiagonals include A002457.
%Y A093375 Antidiagonal sums are A001792.
%Y A093375 See A103283 for a signed version.
%Y A093375 Cf. A103406, A059298, A073107 (unsigned inverse).
%K A093375 nonn,tabl
%O A093375 1,2
%A A093375 _Ralf Stephan_, Apr 28 2004