A093563 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2 - 4\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	4														11
2	14	36	16													66
3	28	128	176	64												396
4	56	368	864	832	256											2376
5	112	960	3200	5120	3840	1024										14256
6	224	2368	10240	23040	28160	17408	4096									85536
7	448	5632	29952	87040	148480	147456	77824	16384								513216
8	896	13056	82432	293888	645120	888832	745472	344064	65536							3079296
9	1792	29696	217088	917504	2465792	4358144	5046272	3670016	1507328	262144						18475776
10	3584	66560	552960	2703360	8601600	18579456	27525120	27525120	17694720	6553600	1048576					110854656
11	7168	147456	1372160	7618560	28016640	71565312	129368064	165150720	145489920	83886080	28311552	4194304				665127936
12	14336	323584	3334144	20725760	86507520	255197184	544997376	847773696	951582720	749731840	392167424	121634816	16777216			3990767616
13	28672	704512	7962624	54788096	255918080	856424448	2110783488	3875536896	5294260224	5305794560	3783262208	1811939328	520093696	67108864		23944605696
14	57344	1523712	18743296	141426688	730988544	2736521216	7647264768	16194207744	26090668032	31788630016	28789702656	18756927488	8287944704	2214592512	268435456	143667634176

Technical details

Node id: n-64a8903e8a3e4c39 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-850d85641d7462e1

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:    7,     4;
  2:   14,    36,     16;
  3:   28,   128,    176,     64;
  4:   56,   368,    864,    832,     256;
  5:  112,   960,   3200,   5120,    3840,    1024;
  6:  224,  2368,  10240,  23040,   28160,   17408,    4096;
  7:  448,  5632,  29952,  87040,  148480,  147456,   77824,   16384;
  8:  896, 13056,  82432, 293888,  645120,  888832,  745472,  344064,   65536;
  9: 1792, 29696, 217088, 917504, 2465792, 4358144, 5046272, 3670016, 1507328, 262144;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (4*n - 4)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (4*n - 4)*x)*v[i - 1])/(n - 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from (6,1)-Pascal triangle.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.