A038328 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 16\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 704\,x + 256\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	704	256								1444
1	10648	23232	16896	4096							54872
2	234256	681472	743424	360448	65536						2085136
3	5153632	18740480	27258880	19824640	7208960	1048576					79235168
4	113379904	494748672	899543040	872284160	475791360	138412032	16777216				3010936384
5	2494357888	12698549248	27705925632	33582940160	24423956480	10657726464	2583691264	268435456			114415582592
6	54875873536	319277809664	812707151872	1182119493632	1074654085120	625253285888	227364831232	47244640256	4294967296		4347792138496
7	1207269217792	7902125789184	22988002295808	39009943289856	42556301770752	30950037651456	15006078861312	4677219385344	850403524608	68719476736	165216101262848

All 15 rows

Technical details

Node id: n-e6ce82432a2d62f1 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-e348a50f4e54dede

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:             484,              704,               256;
  1:           10648,            23232,             16896,              4096;
  2:          234256,           681472,            743424,            360448,             65536;
  3:         5153632,         18740480,          27258880,          19824640,           7208960,           1048576;
  4:       113379904,        494748672,         899543040,         872284160,         475791360,         138412032,          16777216;
  5:      2494357888,      12698549248,       27705925632,       33582940160,       24423956480,       10657726464,        2583691264,         268435456;
  6:     54875873536,     319277809664,      812707151872,     1182119493632,     1074654085120,      625253285888,      227364831232,       47244640256,       4294967296;
  7:   1207269217792,    7902125789184,    22988002295808,    39009943289856,    42556301770752,    30950037651456,    15006078861312,     4677219385344,     850403524608,      68719476736;
  8:  26559922791424,  193163074846720,   632170063134720,  1226026789109760,  1560397731594240,  1361801656664064,   825334337372160,   342996088258560,   93544387706880,   15118284881920,   1099511627776;
  9: 584318301411328, 4674546411290624, 16998350586511360, 37087310370570240, 53945178720829440, 54926000152117248, 39946181928812544, 20751263339642880, 7545913941688320, 1829312470712320, 266081813921792, 17592186044416;

%F Formula

The row polynomials satisfy P(n,x) = (22+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 704*x + 256*x^2.

%t Mathematica

P[0, x_] := 484 + 704*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 16*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] = 484 + 704*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 16*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 basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.