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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	1048576	4718592	9502720	11272192	8716288	4587520	1662976	409600	65536	6144	256							41990400
9	4194304	20971520	47448064	64094208	57409536	35782656	15826944	4964352	1081344	155648	13312	512						251942400
10	16777216	92274688	231735296	351272960	357826560	257949696	134873088	51511296	14254080	2785280	364544	28672	1024					1511654400
11	67108864	402653184	1111490560	1868562432	2133852160	1747451904	1055391744	475791360	160038912	39649280	7028736	843776	61440	2048				9069926400
12	268435456	1744830464	5251268608	9697230848	12272533504	11257511936	7716470784	4013948928	1591738368	478674944	107413504	17432576	1933312	131072	4096			54419558400
13	1073741824	7516192768	24494735360	49291460608	68484595712	69575114752	53380907008	31488737280	14394851328	5098176512	1387003904	284557312	42598400	4390912	278528	8192		326517350400
14	4294967296	32212254720	113011326976	246155313152	372521304064	415269650432	352673857536	232716763136	120556879872	49182408704	15744368640	3912237056	739508224	102760448	9895936	589824	16384	1959104102400

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.