A159854 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 9 + 6\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	6	1															16
1	27	36	15	2														80
2	81	162	117	36	4													400
3	243	648	675	342	84	8												2000
4	729	2430	3321	2376	936	192	16											10000
5	2187	8748	14823	13770	7560	2448	432	32										50000
6	6561	30618	61965	70956	50220	22464	6192	960	64									250000
7	19683	104976	247131	336798	292572	167832	63504	15264	2112	128								1250000
8	59049	354294	951345	1504656	1551312	1088640	526176	172800	36864	4608	256							6250000
9	177147	1180980	3562623	6416658	7663248	6368544	3755808	1570752	456192	87552	9984	512						31250000
10	531441	3897234	13049829	26375220	35823060	34432128	24004512	12223872	4510080	1175040	205056	21504	1024					156250000
11	1594323	12754584	46943955	105225318	160219620	174942504	140877792	84680640	37977984	12545280	2965248	474624	46080	2048				781250000
12	4782969	41452398	166341033	409563864	691109496	845266752	772518384	535797504	283295232	113591808	33986304	7354368	1087488	98304	4096			3906250000
13	14348907	133923132	581927895	1561373658	2892456216	3918019248	4008088656	3152429280	1921480704	907365888	329142528	90035712	17971200	2469888	208896	8192		19531250000
14	43046721	430467210	2013629949	5847976764	11800115964	17538970176	19860304464	17473465152	12069300672	6565059072	2802159360	928392192	233985024	43352064	5566464	442368	16384	97656250000

Technical details

Node id: n-4efc2409ba345227 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-60d64fb2105f3f8c

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:      9,       6,       1;
  1:     27,      36,      15,       2;
  2:     81,     162,     117,      36,       4;
  3:    243,     648,     675,     342,      84,       8;
  4:    729,    2430,    3321,    2376,     936,     192,      16;
  5:   2187,    8748,   14823,   13770,    7560,    2448,     432,      32;
  6:   6561,   30618,   61965,   70956,   50220,   22464,    6192,     960,     64;
  7:  19683,  104976,  247131,  336798,  292572,  167832,   63504,   15264,   2112,   128;
  8:  59049,  354294,  951345, 1504656, 1551312, 1088640,  526176,  172800,  36864,  4608,  256;
  9: 177147, 1180980, 3562623, 6416658, 7663248, 6368544, 3755808, 1570752, 456192, 87552, 9984, 512;

%F Formula

The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 6*x + x^2.

%t Mathematica

P[0, x_] := 9 + 6*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 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] = 9 + 6*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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_A013610 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 A013610

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.