A093645 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 11 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	11	1														12
2	22	13	1													36
3	44	48	15	1												108
4	88	140	78	17	1											324
5	176	368	296	112	19	1										972
6	352	912	960	520	150	21	1									2916
7	704	2176	2832	2000	820	192	23	1								8748
8	1408	5056	7840	6832	3640	1204	238	25	1							26244
9	2816	11520	20736	21504	14112	6048	1680	288	27	1						78732
10	5632	25856	52992	63744	49728	26208	9408	2256	342	29	1					236196
11	11264	57344	131840	180480	163200	102144	45024	13920	2940	400	31	1				708588
12	22528	125952	321024	492800	506880	367488	192192	72864	19800	3740	462	33	1			2125764
13	45056	274432	768000	1306624	1506560	1241856	751872	337920	112464	27280	4664	528	35	1		6377292
14	90112	593920	1810432	3381248	4319744	3990272	2745600	1427712	562848	167024	36608	5720	598	37	1	19131876

Technical details

Node id: n-2f02c090d42e1cb9 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-82653a95da5066d8

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:   11,     1;
  2:   22,    13,     1;
  3:   44,    48,    15,     1;
  4:   88,   140,    78,    17,     1;
  5:  176,   368,   296,   112,    19,    1;
  6:  352,   912,   960,   520,   150,   21,    1;
  7:  704,  2176,  2832,  2000,   820,  192,   23,   1;
  8: 1408,  5056,  7840,  6832,  3640, 1204,  238,  25,  1;
  9: 2816, 11520, 20736, 21504, 14112, 6048, 1680, 288, 27, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 11 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 11 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*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] = 11 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*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 (10,1) 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.