A093562 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 6 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	2														8
2	12	16	4													32
3	24	56	40	8												128
4	48	160	192	96	16											512
5	96	416	704	576	224	32										2048
6	192	1024	2240	2560	1600	512	64									8192
7	384	2432	6528	9600	8320	4224	1152	128								32768
8	768	5632	17920	32256	35840	25088	10752	2560	256							131072
9	1536	12800	47104	100352	136192	121856	71680	26624	5632	512						524288
10	3072	28672	119808	294912	473088	516096	387072	196608	64512	12288	1024					2097152
11	6144	63488	296960	829440	1536000	1978368	1806336	1167360	522240	153600	26624	2048				8388608
12	12288	139264	720896	2252800	4730880	7028736	7569408	5947392	3379200	1351680	360448	57344	4096			33554432
13	24576	303104	1720320	5947392	13967360	23519232	29196288	27033600	18653184	9461760	3424256	835584	122880	8192		134217728
14	49152	655360	4046848	15335424	39829504	74973184	105431040	112459776	91373568	56229888	25772032	8519680	1916928	262144	16384	536870912

Technical details

Node id: n-37e7465556f33a46 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b17361b6ed7801be

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:    6,     2;
  2:   12,    16,     4;
  3:   24,    56,    40,      8;
  4:   48,   160,   192,     96,     16;
  5:   96,   416,   704,    576,    224,     32;
  6:  192,  1024,  2240,   2560,   1600,    512,    64;
  7:  384,  2432,  6528,   9600,   8320,   4224,  1152,   128;
  8:  768,  5632, 17920,  32256,  35840,  25088, 10752,  2560,  256;
  9: 1536, 12800, 47104, 100352, 136192, 121856, 71680, 26624, 5632, 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) = 6 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 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] = 6 + 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 (5,1) 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

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
[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.