A134991 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	3														7
3	26	40	15													81
4	236	550	420	105												1311
5	2752	8596	9940	5040	945											27273
6	39208	153580	238168	182700	69300	10395										693351
7	660032	3109296	6050940	6224680	3568950	1081080	135135									20830113
8	12818912	70564896	165256080	213380860	164023860	75044970	18918900	2027025								722035503
9	282137824	1777121984	4865852376	7562881040	7296949660	4474530060	1702701000	367567200	34459425							28364200569
10	6939897856	49223755328	154288092816	280452180008	325752427000	250698968520	127819451760	41626985400	7856748900	654729075						1245313236663
11	188666182784	1488011230784	5254095662816	10936000675600	14857583060320	13765443371680	8807063265000	3841775617680	1093397555250	183324141000	13749310575					60429110073489
12	5617349020544	48765233950784	191530074540896	449200114963600	698927604094720	757461528984160	583388863817760	319290709537800	121684453991100	30752624652750	4638100767300	316234143225				3211572892464639
13	181790703209728	1722537395565312	7448810764402176	19436704784538496	34082062554780240	42305209109042880	38113671490214320	25109194083573600	12004345849696200	4061430238565700	922982052692700	126493657290000	7905853580625			185523138537151977
14	6353726042486272	65250314121495552	308054663853123840	885203903046541312	1727210528745968448	2416384728803725680	2493497730018174880	1921496815981863120	1105681848120649800	469201095855191700	142714360609821000	29462481010462500	3699939475732500	213458046676875		11574425593731913479
15	238513970965257728	2638884554522186752	13505736026966520576	42377969658534050816	91071826779554626720	141790494656856689280	164917455279260443760	145564738637641042720	97973002265922944160	50035499784720450000	19085776927271554500	5272495973796051000	997133688709908750	115551955934415000	6190283353629375	775591270444009771137

Technical details

Node id: n-c8d7801a5ac304d1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9675e676835fe620

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:          4,           3;
   3:         26,          40,           15;
   4:        236,         550,          420,          105;
   5:       2752,        8596,         9940,         5040,          945;
   6:      39208,      153580,       238168,       182700,        69300,        10395;
   7:     660032,     3109296,      6050940,      6224680,      3568950,      1081080,       135135;
   8:   12818912,    70564896,    165256080,    213380860,    164023860,     75044970,     18918900,     2027025;
   9:  282137824,  1777121984,   4865852376,   7562881040,   7296949660,   4474530060,   1702701000,   367567200,   34459425;
  10: 6939897856, 49223755328, 154288092816, 280452180008, 325752427000, 250698968520, 127819451760, 41626985400, 7856748900, 654729075;

%F Formula

The row polynomials satisfy P(n,x) = (n+2+(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 + 2 + (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 + 2 + (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 of Ward numbers T(n,k) read by rows.?
  - 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