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A288875 Triangle read by rows. The rows give the coefficients of the numerator polynomials for the o.g.f.s of the diagonal sequences of triangle A028338.

%I A288875 #38 Dec 20 2023 08:06:35
%S A288875 1,1,1,3,8,1,15,71,33,1,105,744,718,112,1,945,9129,14542,5270,353,1,
%T A288875 10395,129072,300291,191384,33057,1080,1,135135,2071215,6524739,
%U A288875 6338915,2033885,190125,3265,1,2027025,37237680,150895836,204889344,103829590,18990320,1038780,9824,1
%N A288875 Triangle read by rows. The rows give the coefficients of the  numerator polynomials for the o.g.f.s of the diagonal sequences of triangle A028338.
%C A288875 The Sheffer triangle  A028338 of the type (1/sqrt(1-2*x), -(1/2)*log(1 - 2*x)) is called here |S1hat[2,1]|. The o.g.f. of the sequence of diagonal d, d >= 0 is D(d, t) = Sum_{m=0..d} A028338(d+m, m)*t^m. The e.g.f. of these o.g.f.s is taken as ED(y,t) := Sum_{d >= 0} D(d, t)*y^(d+1)/(d+1)!.
%C A288875 This e.g.f. is found to be ED(y,t) = 1 - sqrt(1 - 2*x(t;y)), where x = x(t;y) is the compositional inverse of y = y(t;x) = x*(1 - t*(-log(1-2*x)/(2*x))) = x + t*log(1-2*x)/2. The o.g.f.s are then D(d, t) = P(d, t)/(1 - t)^(2*d+1), with the row polynomials P(d, t) = Sum_{m=0..d} T(d, m)*t^m, d >= 0.
%C A288875 This computation was inspired by an article of P. Bala (see a link under, e.g., A112007) for Sheffer triangles of the Jabotinsky type (1, F(x)). There Sheffer is called exponential Riordan, and the diagonals are labeled by n = d+1, n >= 1.
%H A288875 Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A288875 Wolfdieter Lang, <a href="http://arxiv.org/abs/1708.01421">On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%F A288875 T(n, m) = [x^m] P(n, x), with the numerator polynomial of the o.g.f. of the diagonal n (main diagonal n=0) D(n, x) = P(n, x)/(1-x)^(2*n+1). See a comment above.
%F A288875 T(n, m) = Sum_{i=0..n-m} ( (-1)^(i-n+m)*binomial(2*n+1,n-m-i)*(1/(2^i*i!))*Sum_{j=0..i} (-1)^(i-j)*binomial(i,j)*(2*j+1)^(n+i) ). - _Detlef Meya_, Dec 18 2023, after _Peter Bala_ from A214406.
%e A288875 The triangle T(n, m) begins:
%e A288875 n\m      0        1         2         3         4        5       6    7  8 ...
%e A288875 0:       1
%e A288875 1:       1        1
%e A288875 2:       3        8         1
%e A288875 3:      15       71        33         1
%e A288875 4:     105      744       718       112         1
%e A288875 5:     945     9129     14542      5270       353        1
%e A288875 6:   10395   129072    300291    191384     33057     1080       1
%e A288875 7:  135135  2071215   6524739   6338915   2033885   190125    3265    1
%e A288875 8: 2027025 37237680 150895836 204889344 103829590 18990320 1038780 9824  1
%e A288875 ...
%t A288875 De[d_, t_] := Sum[A028338[d+m, m] t^m, {m, 0, d}]; A028338[n_, k_] := SeriesCoefficient[Times @@ Table[x+i, {i, 1, 2n-1, 2}], {x, 0, k}]; P[n_, x_] := De[n, x] (1-x)^(2n+1); T[n_, m_] := Coefficient[P[n, x], x, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 24 2017 *)
%t A288875 T[n_,m_]:=Sum[(-1)^(i-n+m)*Binomial[2*n+1,n-m-i]*(1/(2^i*i!)*Sum[(-1)^(i-j)*Binomial[i,j]*(2*j+1)^(n+i),{j,0,i}]),{i,0,n-m}];Flatten[Table[T[n,m],{n,0,8},{m,0,n}]] (* _Detlef Meya_, Dec 18 2023, after _Peter Bala_ from A214406 *)
%Y A288875 Cf. A028338, A112007, A214406.
%K A288875 nonn,tabl
%O A288875 0,4
%A A288875 _Wolfdieter Lang_, Jul 21 2017