A093561 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	10	28	16													54
3	20	96	144	64												324
4	40	272	672	704	256											1944
5	80	704	2432	4096	3328	1024										11664
6	160	1728	7680	17920	23040	15360	4096									69984
7	320	4096	22272	66560	117760	122880	69632	16384								419904
8	640	9472	60928	222208	501760	716800	630784	311296	65536							2519424
9	1280	21504	159744	688128	1892352	3440640	4128768	3145728	1376256	262144						15116544
10	2560	48128	405504	2015232	6537216	14450688	22020096	22806528	15335424	6029312	1048576					90699264
11	5120	106496	1003520	5652480	21135360	55050240	101842944	133693440	121896960	73400320	26214400	4194304				544195584
12	10240	233472	2433024	15319040	64880640	194641920	423886848	674758656	778567680	634388480	346030080	113246208	16777216			3265173504
13	20480	507904	5799936	40370176	191037440	648806400	1626341376	3045064704	4256169984	4383047680	3229614080	1610612736	486539264	67108864		19591041024
14	40960	1097728	13631488	103940096	543555584	2061762560	5847908352	12595494912	20692598784	25790775296	23991418880	16139681792	7415529472	2080374784	268435456	117546246144

Technical details

Node id: n-38c87f97096dd293 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-19b0118601c5a6d7

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:    5,     4;
  2:   10,    28,     16;
  3:   20,    96,    144,     64;
  4:   40,   272,    672,    704,     256;
  5:   80,   704,   2432,   4096,    3328,    1024;
  6:  160,  1728,   7680,  17920,   23040,   15360,    4096;
  7:  320,  4096,  22272,  66560,  117760,  122880,   69632,   16384;
  8:  640,  9472,  60928, 222208,  501760,  716800,  630784,  311296,   65536;
  9: 1280, 21504, 159744, 688128, 1892352, 3440640, 4128768, 3145728, 1376256, 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) = 5 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 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] = 5 + 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 (4,1) 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.