A201637 → reverse (shift +1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	3	4	1						8
3	15	25	11	1					52
4	105	210	130	26	1				472
5	945	2205	1750	546	57	1			5504
6	10395	27720	26775	11368	2037	120	1		78416
7	135135	405405	460845	247555	63805	7071	247	1	1320064

All 15 rows

Technical details

Node id: n-25fb05ce44745668 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-e733eac92d45f846

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:        1,         1;
  2:        3,         4,         1;
  3:       15,        25,        11,         1;
  4:      105,       210,       130,        26,        1;
  5:      945,      2205,      1750,       546,       57,        1;
  6:    10395,     27720,     26775,     11368,     2037,      120,       1;
  7:   135135,    405405,    460845,    247555,    63805,     7071,     247,     1;
  8:  2027025,   6756750,   8828820,   5735730,  1939630,   325930,   23436,   502,    1;
  9: 34459425, 126351225, 186486300, 142101960, 59629570, 13609310, 1561516, 75328, 1013, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1+n*x)*P(n-1,x) - (x+x^2)*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[(2*n - 1 + n*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {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] = (2*n - 1 + n*x)*v[i - 1] - (x + x^2)*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 - 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 Row reversed version of triangle A201637 (second-order Eulerian triangle).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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