A391533 → Pₙ + Pₙ₊₁ → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 27 + 27\,x + 9\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	P_n(1)
0	27	27	9	1															64
1	54	81	45	11	1														192
2	108	216	171	67	13	1													576
3	216	540	558	305	93	15	1												1728
4	432	1296	1656	1168	491	123	17	1											5184
5	864	3024	4608	3992	2150	737	157	19	1										15552
6	1728	6912	12240	12592	8292	3624	1051	195	21	1									46656
7	3456	15552	31392	37424	29176	15540	5726	1441	237	23	1								139968
8	6912	34560	78336	106240	95776	60256	26992	8608	1915	283	25	1							419904
9	13824	76032	191232	290816	297792	216288	114240	44208	12438	2481	333	27	1						1259712
10	27648	165888	458496	772864	886400	730368	444768	202656	69084	17400	3147	387	29	1					3779136
11	55296	359424	1082880	2004224	2545664	2347136	1619904	850080	340824	103884	23694	3921	445	31	1				11337408
12	110592	774144	2525184	5091328	7095552	7239936	5586944	3320064	1531728	548592	151272	31536	4811	507	33	1			34012224
13	221184	1658880	5824512	12707840	19282432	21575424	18413824	12227072	6383520	2628912	851136	214344	41158	5825	573	35	1		102036672
14	442368	3538944	13307904	31240192	51272704	62433280	58403072	42867968	24994112	11641344	4331184	1279824	296660	52808	6971	643	37	1	306110016

Technical details

Node id: n-91056c6e245e4959 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a2815827143c26a7

Observed on the stored rows ?

Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
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:    27,    27,      9,      1;
  1:    54,    81,     45,     11,      1;
  2:   108,   216,    171,     67,     13,      1;
  3:   216,   540,    558,    305,     93,     15,      1;
  4:   432,  1296,   1656,   1168,    491,    123,     17,     1;
  5:   864,  3024,   4608,   3992,   2150,    737,    157,    19,     1;
  6:  1728,  6912,  12240,  12592,   8292,   3624,   1051,   195,    21,    1;
  7:  3456, 15552,  31392,  37424,  29176,  15540,   5726,  1441,   237,   23,   1;
  8:  6912, 34560,  78336, 106240,  95776,  60256,  26992,  8608,  1915,  283,  25,  1;
  9: 13824, 76032, 191232, 290816, 297792, 216288, 114240, 44208, 12438, 2481, 333, 27, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 27*x + 9*x^2 + x^3.

%t Mathematica

P[0, x_] := 27 + 27*x + 9*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(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] = 27 + 27*x + 9*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (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 shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array ((1+x)^3,x).?
    
    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
    
    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 Riordan array ((1+x)^3,x).?
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of Riordan array ((1+x)^3,x).?
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

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.