A093565 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 9 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	9	2														11
2	18	22	4													44
3	36	80	52	8												176
4	72	232	264	120	16											704
5	144	608	992	768	272	32										2816
6	288	1504	3200	3520	2080	608	64									11264
7	576	3584	9408	13440	11200	5376	1344	128								45056
8	1152	8320	25984	45696	49280	33152	13440	2944	256							180224
9	2304	18944	68608	143360	189952	164864	93184	32768	6400	512						720896
10	4608	42496	175104	423936	666624	709632	516096	251904	78336	13824	1024					2883584
11	9216	94208	435200	1198080	2181120	2752512	2451456	1536000	660480	184320	29696	2048				11534336
12	18432	206848	1058816	3266560	6758400	9867264	10407936	7974912	4392960	1689600	428032	63488	4096			46137344
13	36864	450560	2531328	8650752	20049920	33251328	40550400	36765696	24735744	12165120	4235264	983040	135168	8192		184549376
14	73728	974848	5963776	22364160	57401344	106602496	147603456	154632192	123002880	73801728	32800768	10436608	2236416	286720	16384	738197504

Technical details

Node id: n-351271d3c21ad969 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-6c7838f4b8fac103

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:    9,     2;
  2:   18,    22,     4;
  3:   36,    80,    52,      8;
  4:   72,   232,   264,    120,     16;
  5:  144,   608,   992,    768,    272,     32;
  6:  288,  1504,  3200,   3520,   2080,    608,    64;
  7:  576,  3584,  9408,  13440,  11200,   5376,  1344,   128;
  8: 1152,  8320, 25984,  45696,  49280,  33152, 13440,  2944,  256;
  9: 2304, 18944, 68608, 143360, 189952, 164864, 93184, 32768, 6400, 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) = 9 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 9 + 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] = 9 + 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 (8,1) Pascal triangle.?
  - 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
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.