A249250 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5 + 8\,x + 2\,x^{2} - 8\,n - 11\,n x - 2\,n x^{2} + 3\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(30 + 63\,x + 37\,x^{2} + 4\,x^{3} - 12\,n - 24\,n x - 13\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(12 + 36\,x + 37\,x^{2} + 14\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	10	8	1													19
3	28	36	12	1												77
4	76	136	78	16	1											307
5	208	476	388	136	20	1										1229
6	568	1584	1680	848	210	24	1									4915
7	1552	5104	6672	4424	1580	300	28	1								19661
8	4240	16064	24976	20576	9700	2648	406	32	1							78643
9	11584	49680	89568	88320	51984	18756	4116	528	36	1						314573
10	31648	151552	310896	357312	252840	114192	33096	6048	666	40	1					1258291
11	86464	457280	1051840	1381296	1143600	622704	226128	54480	8508	820	44	1				5033165
12	236224	1367424	3485856	5150848	4888800	3123072	1369536	413376	84924	11560	990	48	1			20132659
13	645376	4058560	11357376	18653824	19978240	14667552	7559808	2757504	709200	126700	15268	1176	52	1		80530637
14	1763200	11969792	36479872	65938432	78689600	65337088	38772384	16640640	5172504	1155568	182336	19696	1378	56	1	322122547

Technical details

Node id: n-3b28187da4f2f288 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-3b2315a230a09078

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 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     4,     1;
  2:    10,     8,     1;
  3:    28,    36,    12,     1;
  4:    76,   136,    78,    16,     1;
  5:   208,   476,   388,   136,    20,     1;
  6:   568,  1584,  1680,   848,   210,    24,    1;
  7:  1552,  5104,  6672,  4424,  1580,   300,   28,   1;
  8:  4240, 16064, 24976, 20576,  9700,  2648,  406,  32,  1;
  9: 11584, 49680, 89568, 88320, 51984, 18756, 4116, 528, 36, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (3*n^2-8*n+5+(3*n^2-11*n+8)*x-(2*n-2)*x^2)*P(n-1,x) - (12*n-30+(24*n-63)*x+(13*n-37)*x^2+(n-4)*x^3)*d/dx P(n-1,x) + (12+36*x+37*x^2+14*x^3+x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 5 + (3*n^2 - 11*n + 8)*x - (2*n - 2)*x^2)*P[n - 1, x] - (12*n - 30 + (24*n - 63)*x + (13*n - 37)*x^2 + (n - 4)*x^3)*D[P[n - 1, x], x] + (12 + 36*x + 37*x^2 + 14*x^3 + x^4)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - 8*n + 5 + (3*n^2 - 11*n + 8)*x - (2*n - 2)*x^2)*v[i - 1] - (12*n - 30 + (24*n - 63)*x + (13*n - 37)*x^2 + (n - 4)*x^3)*deriv(v[i - 1], x) + (12 + 36*x + 37*x^2 + 14*x^3 + x^4)*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); 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 ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A249250 → x ↦ x/(x+1)

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