A097806 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	2	6	4													12
3	2	10	16	8												36
4	2	14	36	40	16											108
5	2	18	64	112	96	32										324
6	2	22	100	240	320	224	64									972
7	2	26	144	440	800	864	512	128								2916
8	2	30	196	728	1680	2464	2240	1152	256							8748
9	2	34	256	1120	3136	5824	7168	5632	2560	512						26244
10	2	38	324	1632	5376	12096	18816	19968	13824	5632	1024					78732
11	2	42	400	2280	8640	22848	43008	57600	53760	33280	12288	2048				236196
12	2	46	484	3080	13200	40128	88704	143616	168960	140800	78848	26624	4096			708588
13	2	50	576	4048	19360	66528	168960	321024	456192	478720	360448	184320	57344	8192		2125764
14	2	54	676	5200	27456	105248	302016	658944	1098240	1391104	1317888	905216	425984	122880	16384	6377292

Technical details

Node id: n-dd72b1be3ab7cf4c · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-04668acf2e74f66d

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: 2,  2;
  2: 2,  6,   4;
  3: 2, 10,  16,    8;
  4: 2, 14,  36,   40,   16;
  5: 2, 18,  64,  112,   96,   32;
  6: 2, 22, 100,  240,  320,  224,   64;
  7: 2, 26, 144,  440,  800,  864,  512,  128;
  8: 2, 30, 196,  728, 1680, 2464, 2240, 1152,  256;
  9: 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Riordan array (1+x, 1) read by rows.?
  - 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

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.