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A288875
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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.
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3
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1, 1, 1, 3, 8, 1, 15, 71, 33, 1, 105, 744, 718, 112, 1, 945, 9129, 14542, 5270, 353, 1, 10395, 129072, 300291, 191384, 33057, 1080, 1, 135135, 2071215, 6524739, 6338915, 2033885, 190125, 3265, 1, 2027025, 37237680, 150895836, 204889344, 103829590, 18990320, 1038780, 9824, 1
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OFFSET
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0,4
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COMMENTS
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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)!.
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.
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.
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LINKS
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FORMULA
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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.
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.
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EXAMPLE
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The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7 8 ...
0: 1
1: 1 1
2: 3 8 1
3: 15 71 33 1
4: 105 744 718 112 1
5: 945 9129 14542 5270 353 1
6: 10395 129072 300291 191384 33057 1080 1
7: 135135 2071215 6524739 6338915 2033885 190125 3265 1
8: 2027025 37237680 150895836 204889344 103829590 18990320 1038780 9824 1
...
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MATHEMATICA
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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[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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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