A200536 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(28 + 30\,x + 8\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	28	30	8													66
2	784	1680	1348	480	64											4356
3	21952	70560	94416	67320	26976	5760	512									287496
4	614656	2634240	4936064	5281920	3530256	1509120	402944	61440	4096							18974736
5	17210368	92198400	222154240	317049600	296793280	190418400	84798080	25881600	5181440	614400	32768					1252332576
6	481890304	3097866240	9123953664	16279603200	19598933760	16771910400	10461244480	4791974400	1599912960	379699200	60801024	5898240	262144			82653950016
7	13492928512	101196963840	352261812224	754330429440	1110149870592	1187818329600	952863627520	582187900800	272246750720	96964761600	25892708352	5026775040	670695424	55050240	2097152	5455160701056

All 15 rows

Technical details

Node id: n-86e0f3639d6a7bfe · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4305ea87645379bc

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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:              1;
  1:             28,              30,               8;
  2:            784,            1680,            1348,              480,               64;
  3:          21952,           70560,           94416,            67320,            26976,             5760,              512;
  4:         614656,         2634240,         4936064,          5281920,          3530256,          1509120,           402944,            61440,             4096;
  5:       17210368,        92198400,       222154240,        317049600,        296793280,        190418400,         84798080,         25881600,          5181440,           614400,           32768;
  6:      481890304,      3097866240,      9123953664,      16279603200,      19598933760,      16771910400,      10461244480,       4791974400,       1599912960,        379699200,        60801024,         5898240,          262144;
  7:    13492928512,    101196963840,    352261812224,     754330429440,    1110149870592,    1187818329600,     952863627520,     582187900800,     272246750720,      96964761600,     25892708352,      5026775040,       670695424,       55050240,       2097152;
  8:   377801998336,   3238302842880,  13007183085568,   32498682101760,   56532203757568,   72598052782080,   71195930423296,   54389716684800,   32711455064320,   15539919052800,   5811912687616,   1693249044480,    376724389888,    61876469760,    7075790848,    503316480,    16777216;
  9: 10578455953408, 102006539550720, 464372627668992, 1326085014159360, 2661919632949248, 3988701047439360, 4623685265375232, 4239574402129920, 3117179685731328, 1851579118886400, 890622767351808, 346087706296320, 107841055752192, 26580265205760, 5068211355648, 721378344960, 72175583232, 4529848320, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (28+30*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(28 + 30*x + 8*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (28 + 30*x + 8*x^2)*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 (−∞, -1]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - ↑ Geometric basis map (elementary) (above)

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.