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

A059110 → x ↦ −x−1

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 ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	1														1
3	0	-2	1													-1
4	0	6	-6	1												1
5	0	-24	36	-12	1											1
6	0	120	-240	120	-20	1										-19
7	0	-720	1800	-1200	300	-30	1									151
8	0	5040	-15120	12600	-4200	630	-42	1								-1091
9	0	-40320	141120	-141120	58800	-11760	1176	-56	1							7841
10	0	362880	-1451520	1693440	-846720	211680	-28224	2016	-72	1						-56519
11	0	-3628800	16329600	-21772800	12700800	-3810240	635040	-60480	3240	-90	1					396271
12	0	39916800	-199584000	299376000	-199584000	69854400	-13970880	1663200	-118800	4950	-110	1				-2442439
13	0	-479001600	2634508800	-4390848000	3293136000	-1317254400	307359360	-43908480	3920400	-217800	7260	-132	1			7701409
14	0	6227020800	-37362124800	68497228800	-57081024000	25686460800	-6849722880	1141620480	-122316480	8494200	-377520	10296	-156	1		145269541
15	0	-87178291200	566658892800	-1133317785600	1038874636800	-519437318400	155831195520	-29682132480	3710266560	-309188880	17177160	-624624	14196	-182	1	-4833158329

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 ?

A111596 triangle 55 terms, 10 rows · tabl · catalog b-file?

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

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, +∞)
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.