Real-rooted atlas

A111596: The matrix inverse of the unsigned Lah numbers A271703.

A059110 → x ↦ −x−1

A111596 ↗

Proved for all n all roots real, in [−1, ∞)

Recurrence from A111596

\(P_{n}(x) = \left(2 + x - n\right)\,P_{n-1}(x) + \left(-x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30-21-1
406-611
50-2436-1211
60120-240120-201-19
70-7201800-1200300-301151
805040-1512012600-4200630-421-1091

All 15 rows b-files: A111596, A111884 (row sums) ?

Technical details
Node id
n-92bd113d4a6eac19 · revision 0 · JSON
Definition source
oeis_catalog:A111596
Fingerprint
f6-8f4dc30445e94001

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

Row sums ? A111884

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0,      1;
   3: 0,     -2,        1;
   4: 0,      6,       -6,       1;
   5: 0,    -24,       36,     -12,       1;
   6: 0,    120,     -240,     120,     -20,      1;
   7: 0,   -720,     1800,   -1200,     300,    -30,      1;
   8: 0,   5040,   -15120,   12600,   -4200,    630,    -42,    1;
   9: 0, -40320,   141120, -141120,   58800, -11760,   1176,  -56,   1;
  10: 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2-x)*P(n-1,x) - x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 2 - x)*P[n - 1, x] - x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(n - 2 - x)*v[i - 1] - x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A111596 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 0,      1;
  2: 0,     -2,        1;
  3: 0,      6,       -6,       1;
  4: 0,    -24,       36,     -12,       1;
  5: 0,    120,     -240,     120,     -20,      1;
  6: 0,   -720,     1800,   -1200,     300,    -30,      1;
  7: 0,   5040,   -15120,   12600,   -4200,    630,    -42,    1;
  8: 0, -40320,   141120, -141120,   58800, -11760,   1176,  -56,   1;
  9: 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-1-x)*P(n-1,x) - x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x] - x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(n - 1 - x)*v[i - 1] - x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.