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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	10666233446400	33873902583552	46596710499984	36319087655496	17564704316073	5402354377320	1032742423800	112258692000	5315050000							151573309044625
9	2591894727475200	9105989470407936	14100660663347376	12646468561284216	7246388336556411	2753077867606746	693949467923640	111963740907600	10496769894000	435834100000						49261325439503125
10	777568418242560000	2993578208597376000	5149903135515414336	5218107295383349776	3451209825656629116	1557808582274221311	486245705005373346	103678018532567640	14457368799867600	1190923989294000	44019244100000					19753791501240753125
11	282257335822049280000	1181532236746439808000	2234631379640975275968	2522461130757036517680	1889398256750125041780	986532114095651088045	366559837954405524540	96957096737977601532	17896743135325190880	2196104401697569200	161271712302168000	5370347780200000				9580588878101765265625
12	121935169075125288960000	551349239968659142656000	1136682930333135091378176	1413724758534981190653120	1181976910875824313112560	700144620518089401093540	301401006540172594367805	95036642294195124920124	21790172061467234682300	3543744856155502572000	388104517960684110000	25704388524860760000	778700428129000000			5527999782664718558265625
13	61821130721088521502720000	300263043406881484449792000	670027616473571545580255232	909994550733868429195421760	839595502764989729159098320	555909397451561459583559980	271834895803942703530378935	99421748754986269377029718	27203846425177059220347180	5501007892520269699995000	799205616152502281010000	79009893035420704020000	4764547166287732200000	132379072781930000000		3742455852864014463945828125
14	36350824864000050643599360000	188733432275300751592513536000	453128058037615721237799100416	667072236276808230846218275584	672951082120386041297047898880	492275039746221114879475637280	269353070030675917213824129840	112011462741308638989178124379	35581946202736405888838996286	8593750386561799250005454460	1553631457124164472132895000	203901323486870323322730000	18366502661755065225540000	1016454686554458083400000	26078677338040210000000	2937827844498251354197475078125

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.