A008955 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	36	39	10						85
3	972	1413	660	100					3145
4	46656	84348	55701	16020	1700				204425
5	3499200	7539156	6370623	2649726	544020	44200			20646925
6	377913600	943699248	966976056	521883459	156794022	24902340	1635400		2993804125
7	55553299200	157619469456	189330442632	125065671273	49142894184	11500345080	1485520800	81770000	589779412625

All 15 rows

Technical details

Node id: n-659e4a6217d8f22e · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-8cb4d875493a3ab1

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:               36,               39,                10;
  3:              972,             1413,               660,               100;
  4:            46656,            84348,             55701,             16020,             1700;
  5:          3499200,          7539156,           6370623,           2649726,           544020,            44200;
  6:        377913600,        943699248,         966976056,         521883459,        156794022,         24902340,         1635400;
  7:      55553299200,     157619469456,      189330442632,      125065671273,      49142894184,      11500345080,      1485520800,        81770000;
  8:   10666233446400,   33873902583552,    46596710499984,    36319087655496,   17564704316073,    5402354377320,   1032742423800,    112258692000,     5315050000;
  9: 2591894727475200, 9105989470407936, 14100660663347376, 12646468561284216, 7246388336556411, 2753077867606746, 693949467923640, 111963740907600, 10496769894000, 435834100000;

%F Formula

The row polynomials satisfy P(n,x) = (3*n^2+(n^2+1)*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*n^2 + (n^2 + 1)*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*n^2 + (n^2 + 1)*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 of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    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 of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A008955 → 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.