A104556: Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

Proved for all n all roots real, in [0, ∞)alternating coefficient signs interlacing (weak)

Recurrence from A104556

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

\(P_{1}(x) = 1\)

\(P_{2}(x) = -1 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	1							0
3	0	-3	1						-2
4	0	3	-6	1					-2
5	0	0	15	-10	1				6
6	0	0	-15	45	-15	1			16
7	0	0	0	-105	105	-21	1		-20
8	0	0	0	105	-420	210	-28	1	-132

All 15 rows b-files: A104556 ?

Technical details

Node id: n-6cb9b08f268a058b · revision 0 · JSON
Definition source: oeis_catalog:A104556
Fingerprint: f6-690bd482f7f311e9

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 ?

A104556 triangle 78 terms, 12 rows · tabl · catalog b-file?

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  1;
   2: -1,  1;
   3:  0, -3,   1;
   4:  0,  3,  -6,    1;
   5:  0,  0,  15,  -10,    1;
   6:  0,  0, -15,   45,  -15,     1;
   7:  0,  0,   0, -105,  105,   -21,     1;
   8:  0,  0,   0,  105, -420,   210,   -28,   1;
   9:  0,  0,   0,    0,  945, -1260,   378, -36,   1;
  10:  0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1;

%F Formula

The row polynomials satisfy P(n,x) = x*P(n-1,x) - (n-1)*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = x*v[i - 1] - (n - 1)*x*v[i - 2]); 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

A104556 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:  1;
  1: -1,  1;
  2:  0, -3,   1;
  3:  0,  3,  -6,    1;
  4:  0,  0,  15,  -10,    1;
  5:  0,  0, -15,   45,  -15,     1;
  6:  0,  0,   0, -105,  105,   -21,     1;
  7:  0,  0,   0,  105, -420,   210,   -28,   1;
  8:  0,  0,   0,    0,  945, -1260,   378, -36,   1;
  9:  0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1;

%F Formula

The row polynomials satisfy P(n,x) = x*P(n-1,x) - n*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - n*x*P[n - 2, 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; v[2] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] - n*x*v[i - 2]); 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 [0, +∞)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n(-x) have one sign
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003