A159854 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 16 + 8\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	16	8	1								25
1	64	64	20	2							150
2	256	384	208	48	4						900
3	1024	2048	1600	608	112	8					5400
4	4096	10240	10496	5632	1664	256	16				32400
5	16384	49152	62464	43520	17920	4352	576	32			194400
6	65536	229376	348160	299008	158720	53248	11008	1280	64		1166400
7	262144	1048576	1851392	1892352	1232896	530432	150528	27136	2816	128	6998400

All 15 rows

Technical details

Node id: n-baff0384c99b98b2 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-23ef07238c102b65

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:      16,        8,        1;
  1:      64,       64,       20,        2;
  2:     256,      384,      208,       48,        4;
  3:    1024,     2048,     1600,      608,      112,        8;
  4:    4096,    10240,    10496,     5632,     1664,      256,       16;
  5:   16384,    49152,    62464,    43520,    17920,     4352,      576,      32;
  6:   65536,   229376,   348160,   299008,   158720,    53248,    11008,    1280,      64;
  7:  262144,  1048576,  1851392,  1892352,  1232896,   530432,   150528,   27136,    2816,    128;
  8: 1048576,  4718592,  9502720, 11272192,  8716288,  4587520,  1662976,  409600,   65536,   6144,   256;
  9: 4194304, 20971520, 47448064, 64094208, 57409536, 35782656, 15826944, 4964352, 1081344, 155648, 13312, 512;

%F Formula

The row polynomials satisfy P(n,x) = (4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 8*x + x^2.

%t Mathematica

P[0, x_] := 16 + 8*x + 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] = 16 + 8*x + 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 (−∞, 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_A013611 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

Related ?

Same fingerprint ?

A159854 → basis A013611

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.