A028313 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 8\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 7 + 28\,x + 16\,x^{2}\)

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	4														6
2	7	28	16													51
3	14	84	144	64												306
4	28	224	624	704	256											1836
5	56	560	2144	3904	3328	1024										11016
6	112	1344	6528	16384	22272	15360	4096									66096
7	224	3136	18432	58880	110080	119808	69632	16384								396576
8	448	7168	49408	191488	455680	679936	618496	311296	65536							2379456
9	896	16128	127488	580608	1677312	3182592	3956736	3096576	1376256	262144						14276736
10	1792	35840	319488	1671168	5677056	13074432	20643840	22020096	15138816	6029312	1048576					85660416
11	3584	78848	782336	4620288	18038784	48857088	93585408	126615552	118358016	72613888	26214400	4194304				513962496
12	7168	172032	1880064	12369920	54558720	169869312	382599168	627572736	743178240	618659840	342884352	113246208	16777216			3083774976
13	14336	372736	4448256	32260096	158597120	557973504	1444675584	2785542144	3996647424	4210032640	3160408064	1598029824	486539264	67108864		18502649856
14	28672	802816	10387456	82313216	446234624	1750335488	5121245184	11349786624	19135463424	24406654976	23160946688	15837691904	7365197824	2080374784	268435456	111015899136

Technical details

Node id: n-9e00e84a547b9ff5 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-7b2b9276cd76a7a4

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,     4;
  2:   7,    28,     16;
  3:  14,    84,    144,     64;
  4:  28,   224,    624,    704,     256;
  5:  56,   560,   2144,   3904,    3328,    1024;
  6: 112,  1344,   6528,  16384,   22272,   15360,    4096;
  7: 224,  3136,  18432,  58880,  110080,  119808,   69632,   16384;
  8: 448,  7168,  49408, 191488,  455680,  679936,  618496,  311296,   65536;
  9: 896, 16128, 127488, 580608, 1677312, 3182592, 3956736, 3096576, 1376256, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 7 + 28*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 7 + 28*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (4*n - 8)*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 + 4*x; v[3] = 7 + 28*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (4*n - 8)*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 the 5-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.