A159854 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 9 + 12\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	12	4															25
1	36	66	40	8														150
2	144	336	292	112	16													900
3	576	1632	1840	1032	288	32												5400
4	2304	7680	10624	7808	3216	704	64											32400
5	9216	35328	57856	52480	28480	9248	1664	128										194400
6	36864	159744	302080	325632	218880	93952	25152	3840	256									1166400
7	147456	712704	1527808	1906688	1526784	813568	288512	65664	8704	512								6998400
8	589824	3145728	7536640	10682368	9920512	6307840	2781184	839680	166144	19456	1024							41990400
9	2359296	13762560	36438016	57802752	61046784	45072384	23740416	8921088	2343936	410112	43008	2048						251942400
10	9437184	59768832	173277184	304087040	359792640	302383104	185106432	83165184	27217920	6328320	992256	94208	4096					1511654400
11	37748736	257949696	812646400	1562902528	2047344640	1929117696	1345191936	702873600	275202048	79749120	16625664	2361344	204800	8192				9069926400
12	150994944	1107296256	3766484992	7876902912	11315183616	11811160064	9239003136	5501878272	2506555392	869400576	226000896	42696704	5541888	442368	16384			54419558400
13	603979776	4731174912	17280532480	39040581632	61014540288	69875007488	60578332672	40485519360	21029978112	8490713088	2642804736	622788608	107560960	12853248	950272	32768		326517350400
14	2415919104	20132659200	78584479744	190723391488	322139324416	401529110528	382063345664	283098742784	165090951168	76022808576	27552645120	7776763904	1675821056	266534912	29507584	2031616	65536	1959104102400

Technical details

Node id: n-289f8b8e041a3cc0 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-efbd87188225e7d4

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:       9,       12,        4;
  1:      36,       66,       40,        8;
  2:     144,      336,      292,      112,       16;
  3:     576,     1632,     1840,     1032,      288,       32;
  4:    2304,     7680,    10624,     7808,     3216,      704,       64;
  5:    9216,    35328,    57856,    52480,    28480,     9248,     1664,     128;
  6:   36864,   159744,   302080,   325632,   218880,    93952,    25152,    3840,     256;
  7:  147456,   712704,  1527808,  1906688,  1526784,   813568,   288512,   65664,    8704,    512;
  8:  589824,  3145728,  7536640, 10682368,  9920512,  6307840,  2781184,  839680,  166144,  19456,  1024;
  9: 2359296, 13762560, 36438016, 57802752, 61046784, 45072384, 23740416, 8921088, 2343936, 410112, 43008, 2048;

%F Formula

The row polynomials satisfy P(n,x) = (4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 12*x + 4*x^2.

%t Mathematica

P[0, x_] := 9 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 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] = 9 + 12*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 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 (−∞, -1]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Riordan array (1-x,x/(1-x)).?
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.