A028262 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 5 + 20\,x + 16\,x^{2}\)

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	5	20	16													41
3	10	60	112	64												246
4	20	160	464	576	256											1476
5	40	400	1568	3008	2816	1024										8856
6	80	960	4736	12288	17664	13312	4096									53136
7	160	2240	13312	43520	84480	97280	61440	16384								318816
8	320	5120	35584	140288	343040	532480	512000	278528	65536							1912896
9	640	11520	91648	422912	1247232	2437120	3153920	2605056	1245184	262144						11477376
10	1280	25600	229376	1212416	4186112	9863168	16056320	17825792	12910592	5505024	1048576					68864256
11	2560	56320	561152	3342336	13221888	36470784	71565312	99876864	97124352	62652416	24117248	4194304				413185536
12	5120	122880	1347584	8929280	39813120	125829120	289013760	486014976	593756160	513802240	298844160	104857600	16777216			2479113216
13	10240	266240	3186688	23248896	115343360	410910720	1081344000	2128084992	3131572224	3402629120	2652897280	1405091840	452984832	67108864		14874679296
14	20480	573440	7438336	59244544	323682304	1283194880	3806330880	8581545984	14775484416	19331547136	18916311040	13421772800	6526337024	1946157056	268435456	89248075776

Technical details

Node id: n-5fe98d2f798725af · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-00a90a62643051ac

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:   5,    20,    16;
  3:  10,    60,   112,     64;
  4:  20,   160,   464,    576,     256;
  5:  40,   400,  1568,   3008,    2816,    1024;
  6:  80,   960,  4736,  12288,   17664,   13312,    4096;
  7: 160,  2240, 13312,  43520,   84480,   97280,   61440,   16384;
  8: 320,  5120, 35584, 140288,  343040,  532480,  512000,  278528,   65536;
  9: 640, 11520, 91648, 422912, 1247232, 2437120, 3153920, 2605056, 1245184, 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) = 5 + 20*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 5 + 20*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] = 5 + 20*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 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.