A176740 -id:A176740 - cp's OEIS frontend

A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

1, 0, 1, 0, -1, 1, 0, 3, -1, -3, 1, 0, -15, 10, -1, 15, -4, -6, 1, 0, 105, -105, 10, 15, -1, -105, 60, -5, 45, -10, -10, 1, 0, -945, 1260, -280, -210, 35, 21, -1, 945, -840, 70, 105, -6, -420, 210, -15, 105, -20, -15, 1

Offset: 0

Peter Luschny, Feb 06 2016

The triangle of coefficients of the Bell polynomials is A268441. For the definition of the inverse Bell polynomials see the link 'Bell transform'.

			[[1]],
[[0], [1]],
[[0], [-1],                    [1]],
[[0], [3, -1],                 [-3],           [1]],
[[0], [-15, 10, -1],           [15, -4],       [-6],      [1]],
[[0], [105, -105, 10, 15, -1], [-105, 60, -5], [45, -10], [-10], [1]]
Replacing the sublists by their sums reduces the triangle to the triangle of the Stirling numbers of first kind (A048994). The column 1 of sublists is A176740 (missing the leading 1) and A134685 in different order.

Peter Luschny, First 21 rows, flattened
Peter Luschny, The Bell transform
Peter Luschny, SageMath implementation

Cf. A036040, A048994, A134685, A176740, A268441.

A268442Matrix[dim_] := Module[ {v, r, A},
v = Table[Subscript[x,j],{j,1,dim}];
r = Table[Subscript[x,j]->1,{j,1,n}];
A = Table[Table[BellY[n,k,v], {k,0,dim}], {n,0,dim}];
Table[Table[MonomialList[Inverse[A][[n,k]]/. r[[1]],
v, Lexicographic] /. r, {k,1,n}] // Flatten, {n,1,dim}]];
A268442Matrix[7] // Flatten

Sage
```
# see link
```

A277394 Lagrange inversion, or reversion, for divided power series with odd powers only.

Original entry on oeis.org

1, -1, 10, -1, -280, 56, -1, 15400, -4620, 126, 120, -1, -1401400, 560560, -36036, -17160, 792, 220, -1, 190590400, -95295200, 10090080, 3203200, -126126, -360360, -50050, 1716, 2002, 364, -1

Offset: 1

Tom Copeland, Oct 12 2016

Coefficients for partition polynomials for compositional inversion order-by-order of odd functions, e.g.f.s, or formal Taylor series f(x) = a1 x + a3 x^3/3! + a5 x^5/5! + ... .
The compositional inverse of f(x) is g(x)
= a1^(-1) [1] x
+ a1^(-4) [-1 a3] x^3/3!
+ a1^(-7) [10 a3^2 - 1 a1 a5] x^5/5!
+ a1^(-10)[-280 a3^3 + 56 a1 a3 a5 - a1^2 a7] x^7/7!
+ a1^(-13)[15400 a3^4 - 4620 a1 a3^2 a5 + a1^2 (126 a5^2 + 120 a3 a7) - a1^3 a9] * x^9/9! ... .

Cf. A133437, A134264, A134685, A133932, A145271, A176740 for other inversion formulas.

rows[nn_] := With[{s = InverseSeries[x + Sum[a[k] x^(2k+1)/(2k+1)!, {k, nn}] + O[x]^(2nn+2)]}, Table[(2n-1)! Coefficient[s, x^(2n-1) Product[a[w], {w, p}]], {n, nn}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n-1]]]}]];
rows[5] // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

Corrected and extended by Andrey Zabolotskiy, Mar 07 2024

cp's OEIS Frontend

A268442 Triangle read by rows, the coefficients of the inverse Bell polynomials.

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