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	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

All 15 rows

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.