A159854 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 8\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 25 + 80\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	25	80	64															169
1	150	680	1024	512														2366
2	900	5280	11584	11264	4096													33124
3	5400	38880	111744	160256	114688	32768												463736
4	32400	276480	981504	1855488	1970176	1114112	262144											6492304
5	194400	1918080	8100864	18984960	26664960	22446080	10485760	2097152										90892256
6	1166400	13063680	63949824	178716672	311869440	347996160	242483200	96468992	16777216									1272491584
7	6998400	87713280	488208384	1583898624	3300950016	4582932480	4238868480	2518679552	872415232	134217728								17814882176
8	41990400	582266880	3630956544	13409058816	32476889088	53905195008	62096670720	49023025152	25383927808	7784628224	1073741824							249408350464
9	251942400	3829524480	26443874304	109502005248	302133805056	583246282752	803821584384	790911516672	544487768064	249779191808	68719476736	8589934592						3491716906496
10	1511654400	24992686080	189299441664	868563025920	2688818872320	5916548136960	9488899768320	11176041775104	9594218741760	5854577295360	2410550394880	601295421440	68719476736					48884036690944
11	9069926400	162049351680	1335738138624	6725773688832	23081417441280	57009839800320	104265783705600	142967448797184	146973646651392	111881213706240	61299920732160	22892175687680	5222680231936	549755813888				684376513673216
12	54419558400	1044855521280	9310823645184	51040547241984	192294694158336	526710378332160	1081673420636160	1691930962427904	2025581470285824	1847076455448576	1262849234042880	627752419983360	214473486893056	45079976738816	4398046511104			9581271191425024
13	326517350400	6704489594880	64223786041344	380729872613376	1562092542885888	4698619823259648	10703723550474240	18804973139656704	25688936521138176	27287110494978048	22353707047845888	13869308392243200	6308860281225216	1986267755577344	387028092977152	35184372088832		134137796679950336
14	1959104102400	42839076372480	438978633007104	2798169524011008	12418394238222336	40688459282644992	101811299888922624	198459627241734144	304573404244082688	369234155138973696	352419126246899712	262045506736226304	148807628825296896	62388488783265792	18212310602481664	3307330976350208	281474976710656	1877929153519304704

Technical details

Node id: n-2f46ab61d8cae267 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-249fd621b3e0dd99

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:        25,         80,          64;
  1:       150,        680,        1024,          512;
  2:       900,       5280,       11584,        11264,         4096;
  3:      5400,      38880,      111744,       160256,       114688,        32768;
  4:     32400,     276480,      981504,      1855488,      1970176,      1114112,       262144;
  5:    194400,    1918080,     8100864,     18984960,     26664960,     22446080,     10485760,      2097152;
  6:   1166400,   13063680,    63949824,    178716672,    311869440,    347996160,    242483200,     96468992,     16777216;
  7:   6998400,   87713280,   488208384,   1583898624,   3300950016,   4582932480,   4238868480,   2518679552,    872415232,    134217728;
  8:  41990400,  582266880,  3630956544,  13409058816,  32476889088,  53905195008,  62096670720,  49023025152,  25383927808,   7784628224,  1073741824;
  9: 251942400, 3829524480, 26443874304, 109502005248, 302133805056, 583246282752, 803821584384, 790911516672, 544487768064, 249779191808, 68719476736, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = (6+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 80*x + 64*x^2.

%t Mathematica

P[0, x_] := 25 + 80*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 8*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] = 25 + 80*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 8*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 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

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.