A290319 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 - 2\,x + 4\,n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	9	24	12													45
3	27	162	276	120												585
4	81	864	3096	4224	1680											9945
5	243	4050	24840	68400	81072	30240										208845
6	729	17496	163620	751680	1748016	1874304	665280									5221125
7	2187	71442	945756	6509160	24787728	51071328	50727744	17297280								151412625
8	6561	279936	4980528	47900160	269637984	896845824	1684323072	1573724160	518918400							4996616625
9	19683	1062882	24459408	313038432	2437519392	11858228928	35545727232	61988156928	55063376640	17643225600						184874815125
10	59049	3936600	113767740	1868572800	19208018592	128200423680	557249880960	1536702105600	2520740093184	2145337989120	670442572800					7579867420125
11	177147	14289858	506640420	10383963480	136104113376	1191338051904	7056167437440	28014601317120	72103708714752	112307097881088	92115523261440	28158588057600				341094033905625
12	531441	51018336	2177254728	54457349760	885974660208	9834803371008	75970052699904	408627506073600	1504982786731776	3653691894521856	5442473072314368	4321789834199040	1295295050649600			16713607661375625
13	1594323	179627058	9082680984	272234785680	5380791468624	73803143123424	719650326650112	5024385153216000	24946323663875328	86210215020154368	199012013943035904	285089023118315520	219975376861900800	64764752532480000		885821206052908125
14	4782969	624974616	36947904084	1307169130176	30843052832592	511972168675968	6144320708615232	53934273098754048	346155769265289984	1605732122909730816	5252387652917443584	11601915822278885376	16054733378974740480	12072964608140083200	3497296636753920000	50491808745015763125

Technical details

Node id: n-7aee7a61dce7c4f1 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-a60199117d78e488

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:     3,       2;
  2:     9,      24,       12;
  3:    27,     162,      276,       120;
  4:    81,     864,     3096,      4224,       1680;
  5:   243,    4050,    24840,     68400,      81072,       30240;
  6:   729,   17496,   163620,    751680,    1748016,     1874304,      665280;
  7:  2187,   71442,   945756,   6509160,   24787728,    51071328,    50727744,    17297280;
  8:  6561,  279936,  4980528,  47900160,  269637984,   896845824,  1684323072,  1573724160,   518918400;
  9: 19683, 1062882, 24459408, 313038432, 2437519392, 11858228928, 35545727232, 61988156928, 55063376640, 17643225600;

%F Formula

The row polynomials satisfy P(n,x) = (3+(4*n-2)*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[(3 + (4*n - 2)*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] = (3 + (4*n - 2)*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 basis_map_A013610 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
  - Geometric basis map (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
  - Geometric basis map (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 (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A290319 → basis A013610

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.