A028262 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 5 + 10\,x + 4\,x^{2}\)

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	5	10	4													19
3	10	30	28	8												76
4	20	80	116	72	16											304
5	40	200	392	376	176	32										1216
6	80	480	1184	1536	1104	416	64									4864
7	160	1120	3328	5440	5280	3040	960	128								19456
8	320	2560	8896	17536	21440	16640	8000	2176	256							77824
9	640	5760	22912	52864	77952	76160	49280	20352	4864	512						311296
10	1280	12800	57344	151552	261632	308224	250880	139264	50432	10752	1024					1245184
11	2560	28160	140288	417792	826368	1139712	1118208	780288	379392	122368	23552	2048				4980736
12	5120	61440	336896	1116160	2488320	3932160	4515840	3796992	2319360	1003520	291840	51200	4096			19922944
13	10240	133120	796672	2906112	7208960	12840960	16896000	16625664	12232704	6645760	2590720	686080	110592	8192		79691776
14	20480	286720	1859584	7405568	20230144	40099840	59473920	67043328	57716736	37756928	18472960	6553600	1593344	237568	16384	318767104

Technical details

Node id: n-20ea15e8be46fca1 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-fcee9e154cda4641

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   2,    2;
  2:   5,   10,     4;
  3:  10,   30,    28,     8;
  4:  20,   80,   116,    72,    16;
  5:  40,  200,   392,   376,   176,    32;
  6:  80,  480,  1184,  1536,  1104,   416,    64;
  7: 160, 1120,  3328,  5440,  5280,  3040,   960,   128;
  8: 320, 2560,  8896, 17536, 21440, 16640,  8000,  2176,  256;
  9: 640, 5760, 22912, 52864, 77952, 76160, 49280, 20352, 4864, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 5 + 10*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 5 + 10*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];
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; v[3] = 5 + 10*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Elements in 3-Pascal triangle (by row).?
  - 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.