A028313 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 7 + 14\,x + 4\,x^{2}\)

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	7	14	4													25
3	14	42	36	8												100
4	28	112	156	88	16											400
5	56	280	536	488	208	32										1600
6	112	672	1632	2048	1392	480	64									6400
7	224	1568	4608	7360	6880	3744	1088	128								25600
8	448	3584	12352	23936	28480	21248	9664	2432	256							102400
9	896	8064	31872	72576	104832	99456	61824	24192	5376	512						409600
10	1792	17920	79872	208896	354816	408576	322560	172032	59136	11776	1024					1638400
11	3584	39424	195584	577536	1127424	1526784	1462272	989184	462336	141824	25600	2048				6553600
12	7168	86016	470016	1546240	3409920	5308416	5978112	4902912	2903040	1208320	334848	55296	4096			26214400
13	14336	186368	1112064	4032512	9912320	17436672	22573056	21762048	15611904	8222720	3086336	780288	118784	8192		104857600
14	28672	401408	2596864	10289152	27889664	54697984	80019456	88670208	74747904	47669248	22618112	7733248	1798144	253952	16384	419430400

Technical details

Node id: n-a39d9e9bb502721f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-ec819eb724397ec0

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:   7,   14,     4;
  3:  14,   42,    36,     8;
  4:  28,  112,   156,    88,     16;
  5:  56,  280,   536,   488,    208,    32;
  6: 112,  672,  1632,  2048,   1392,   480,    64;
  7: 224, 1568,  4608,  7360,   6880,  3744,  1088,   128;
  8: 448, 3584, 12352, 23936,  28480, 21248,  9664,  2432,  256;
  9: 896, 8064, 31872, 72576, 104832, 99456, 61824, 24192, 5376, 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) = 7 + 14*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 7 + 14*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] = 7 + 14*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 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.