A159854 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 4\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 4 + 16\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	4	16	16								36
1	12	64	112	64							252
2	36	240	592	640	256						1764
3	108	864	2736	4288	3328	1024					12348
4	324	3024	11664	23808	27136	16384	4096				86436
5	972	10368	47088	118080	176640	157696	77824	16384			605052
6	2916	34992	182736	542592	1002240	1179648	864256	360448	65536		4235364
7	8748	116640	688176	2358720	5177088	7547904	7311360	4538368	1638400	262144	29647548

All 15 rows

Technical details

Node id: n-e2695402b3f168fd · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-201a8fcec63b5aa2

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:     4,      16,      16;
  1:    12,      64,     112,       64;
  2:    36,     240,     592,      640,       256;
  3:   108,     864,    2736,     4288,      3328,      1024;
  4:   324,    3024,   11664,    23808,     27136,     16384,      4096;
  5:   972,   10368,   47088,   118080,    176640,    157696,     77824,     16384;
  6:  2916,   34992,  182736,   542592,   1002240,   1179648,    864256,    360448,     65536;
  7:  8748,  116640,  688176,  2358720,   5177088,   7547904,   7311360,   4538368,   1638400,    262144;
  8: 26244,  384912, 2531088,  9828864,  24966144,  43352064,  52125696,  42860544,  23068672,   7340032,  1048576;
  9: 78732, 1259712, 9132912, 39610944, 114213888, 229920768, 329785344, 337084416, 240648192, 114294784, 32505856, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 16*x^2.

%t Mathematica

P[0, x_] := 4 + 16*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 4 + 16*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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,x/(1-x)).?
  - Affine, reversal and Möbius substitutions (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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A024462 → basis A007318

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.