A093563 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 7 + 2\,x\)

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	2														9
2	14	18	4													36
3	28	64	44	8												144
4	56	184	216	104	16											576
5	112	480	800	640	240	32										2304
6	224	1184	2560	2880	1760	544	64									9216
7	448	2816	7488	10880	9280	4608	1216	128								36864
8	896	6528	20608	36736	40320	27776	11648	2688	256							147456
9	1792	14848	54272	114688	154112	136192	78848	28672	5888	512						589824
10	3584	33280	138240	337920	537600	580608	430080	215040	69120	12800	1024					2359296
11	7168	73728	343040	952320	1751040	2236416	2021376	1290240	568320	163840	27648	2048				9437184
12	14336	161792	833536	2590720	5406720	7974912	8515584	6623232	3717120	1464320	382976	59392	4096			37748736
13	28672	352256	1990656	6848512	15994880	26763264	32980992	30277632	20680704	10362880	3694592	884736	126976	8192		150994944
14	57344	761856	4685824	17678336	45686784	85516288	119488512	126517248	101916672	62087168	28114944	9158656	2023424	270336	16384	603979776

Technical details

Node id: n-e34531ee3663900f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c34b0be58861906b

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,     2;
  2:   14,    18,     4;
  3:   28,    64,    44,      8;
  4:   56,   184,   216,    104,     16;
  5:  112,   480,   800,    640,    240,     32;
  6:  224,  1184,  2560,   2880,   1760,    544,    64;
  7:  448,  2816,  7488,  10880,   9280,   4608,  1216,   128;
  8:  896,  6528, 20608,  36736,  40320,  27776, 11648,  2688,  256;
  9: 1792, 14848, 54272, 114688, 154112, 136192, 78848, 28672, 5888, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (2*n - 2)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (2*n - 2)*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.