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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	12818912	83383808	235820976	378636940	377404720	239068830	93963870	20945925	2027025							1444071006
10	282137824	2059259808	6642974360	12428733416	14859830700	11771479720	6177231060	2070268200	402026625	34459425						56728401138
11	6939897856	56163653184	203511848144	434740272824	606204607008	576451395520	378518420280	169446437160	49483734300	8511477975	654729075					2490626473326
12	188666182784	1676677413568	6742106893600	16190096338416	25793583735920	28623026432000	22572506636680	12648838882680	4935173172930	1276721696250	197073451575	13749310575				120858220146978
13	5617349020544	54382582971328	240295308491680	640730189504496	1148127719058320	1456389133078880	1340850392801920	902679573355560	440975163528900	152437078643850	35390725420050	4954334910525	316234143225			6423145784929278
14	181790703209728	1904328098775040	9171348159967488	26885515548940672	53518767339318736	76387271663823120	80418880599257200	63222865573787920	37113539933269800	16065776088261900	4984412291258400	1049475709982700	134399510870625	7905853580625		371046277074303954
15	6353726042486272	71604040163981824	373304977974619392	1193258566899665152	2612414431792509760	4143595257549694128	4909882458821900560	4414994546000038000	3027178664102512920	1574882943975841500	611915456465012700	172176841620283500	33162420486195000	3913397522409375	213458046676875	23148851187463826958

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