A290319 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 4\,n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	6	5													12
3	1	15	59	45												120
4	1	28	254	812	585											1680
5	1	45	730	5130	14389	9945										30240
6	1	66	1675	20460	122119	312114	208845									665280
7	1	91	3325	62335	633619	3365089	8011695	5221125								17297280
8	1	120	5964	158760	2441334	21740040	105599276	237560280	151412625							518918400
9	1	153	9924	355572	7680414	102304062	823020596	3722336388	7990901865	4996616625						17643225600
10	1	190	15585	722760	20836578	386479380	4608270890	34174098440	145717348221	300659985630	184874815125					670442572800
11	1	231	23375	1361745	50469738	1240779078	20453925470	223113204930	1546855384261	6275071262691	12511934225955	7579867420125				28158588057600
12	1	276	33770	2413620	111748263	3511917288	76288983980	1143539851080	11586949606111	75883563554436	294890141047050	570616907588100	341094033905625			1295295050649600
13	1	325	47294	4068350	230015643	8987582175	248372931092	4881700066100	67620402309031	643644094253875	4013184755214414	15020233818893550	28301322505722525	16713607661375625		64764752532480000
14	1	378	64519	6574932	445638193	21178411254	724714786367	18045465413976	326350505812331	4227525416632518	38126321750669789	227719025845257492	824373714907080675	1516683700464669450	885821206052908125	3497296636753920000

Technical details

Node id: n-a923a628a898e678 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-678d8eaa2c857cae

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   6,    5;
  3: 1,  15,   59,     45;
  4: 1,  28,  254,    812,     585;
  5: 1,  45,  730,   5130,   14389,      9945;
  6: 1,  66, 1675,  20460,  122119,    312114,    208845;
  7: 1,  91, 3325,  62335,  633619,   3365089,   8011695,    5221125;
  8: 1, 120, 5964, 158760, 2441334,  21740040, 105599276,  237560280,  151412625;
  9: 1, 153, 9924, 355572, 7680414, 102304062, 823020596, 3722336388, 7990901865, 4996616625;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-3)*x)*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[(1 + (4*n - 3)*x)*P[n - 1, 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] = (1 + (4*n - 3)*x)*v[i - 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 (−∞, 0]
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 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A290319: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized …

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.