A269939 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−2, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1 - x + n + n x\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	4	7	3						14
4	26	66	55	15					162
5	236	786	970	525	105				2622
6	2752	11348	18536	14980	5985	945			54546
7	39208	192788	391748	420868	252000	79695	10395		1386702
8	660032	3769328	9160236	12275620	9793630	4650030	1216215	135135	41660226

All 15 rows

Technical details

Node id: n-24581a6190713608 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-49369fb71b7b1723

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): 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         1,          1;
   3:         4,          7,          3;
   4:        26,         66,         55,          15;
   5:       236,        786,        970,         525,         105;
   6:      2752,      11348,      18536,       14980,        5985,         945;
   7:     39208,     192788,     391748,      420868,      252000,       79695,      10395;
   8:    660032,    3769328,    9160236,    12275620,     9793630,     4650030,    1216215,     135135;
   9:  12818912,   83383808,  235820976,   378636940,   377404720,   239068830,   93963870,   20945925,   2027025;
  10: 282137824, 2059259808, 6642974360, 12428733416, 14859830700, 11771479720, 6177231060, 2070268200, 402026625, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (n-1+(n-1)*x)*P(n-1,x) + (2+3*x+x^2)*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 - 1 + (n - 1)*x)*P[n - 1, x] + (2 + 3*x + x^2)*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 - 1 + (n - 1)*x)*v[i - 1] + (2 + 3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (-2, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741