A391533 → Pₙ + Pₙ₊₁ → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 27 + 108\,x + 144\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	P_n(1)
0	27	108	144	64								343
1	54	324	720	704	256							2058
2	108	864	2736	4288	3328	1024						12348
3	216	2160	8928	19520	23808	15360	4096					74088
4	432	5184	26496	74752	125696	125952	69632	16384				444528
5	864	12096	73728	255488	550400	754688	643072	311296	65536			2667168
6	1728	27648	195840	805888	2122752	3710976	4304896	3194880	1376256	262144		16003008
7	3456	62208	502272	2395136	7469056	15912960	23453696	23609344	15532032	6029312	1048576	96018048

All 15 rows

Technical details

Node id: n-3eb516a297f686ad · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-f6df6c2d28e8f6fb

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,    108,     144,       64;
  1:    54,    324,     720,      704,      256;
  2:   108,    864,    2736,     4288,     3328,      1024;
  3:   216,   2160,    8928,    19520,    23808,     15360,      4096;
  4:   432,   5184,   26496,    74752,   125696,    125952,     69632,     16384;
  5:   864,  12096,   73728,   255488,   550400,    754688,    643072,    311296,     65536;
  6:  1728,  27648,  195840,   805888,  2122752,   3710976,   4304896,   3194880,   1376256,    262144;
  7:  3456,  62208,  502272,  2395136,  7469056,  15912960,  23453696,  23609344,  15532032,   6029312,   1048576;
  8:  6912, 138240, 1253376,  6799360, 24518656,  61702144, 110559232, 141033472, 125501440,  74186752,  26214400,   4194304;
  9: 13824, 304128, 3059712, 18612224, 76234752, 221478912, 467927040, 724303872, 815136768, 650379264, 349175808, 113246208, 16777216;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 108*x + 144*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 27 + 108*x + 144*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[(2 + 4*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 + 108*x + 144*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (2 + 4*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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)

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.