A028262 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 5 + 15\,x + 9\,x^{2}\)

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	3														5
2	5	15	9													29
3	10	45	63	27												145
4	20	120	261	243	81											725
5	40	300	882	1269	891	243										3625
6	80	720	2664	5184	5589	3159	729									18125
7	160	1680	7488	18360	26730	23085	10935	2187								90625
8	320	3840	20016	59184	108540	126360	91125	37179	6561							453125
9	640	8640	51552	178416	394632	578340	561330	347733	124659	19683						2265625
10	1280	19200	129024	511488	1324512	2340576	2857680	2379456	1292517	413343	59049					11328125
11	2560	42240	315648	1410048	4183488	8654688	12737088	13331952	9723402	4704237	1358127	177147				56640625
12	5120	92160	758016	3767040	12597120	29859840	51438240	64875168	59442660	38578680	16828965	4428675	531441			283203125
13	10240	199680	1792512	9808128	36495360	97511040	192456000	284065056	313510824	255485340	149393970	59344245	14348907	1594323		1416015625
14	20480	430080	4184064	24993792	102415104	304508160	677445120	1145498112	1479216816	1451503152	1065243960	566870400	206730549	46235367	4782969	7080078125

Technical details

Node id: n-907f7f12164f916f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cf4a3c9ee6bb033d

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,    3;
  2:   5,   15,     9;
  3:  10,   45,    63,     27;
  4:  20,  120,   261,    243,     81;
  5:  40,  300,   882,   1269,    891,    243;
  6:  80,  720,  2664,   5184,   5589,   3159,    729;
  7: 160, 1680,  7488,  18360,  26730,  23085,  10935,   2187;
  8: 320, 3840, 20016,  59184, 108540, 126360,  91125,  37179,   6561;
  9: 640, 8640, 51552, 178416, 394632, 578340, 561330, 347733, 124659, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 5 + 15*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 5 + 15*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (3*n - 6)*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 + 3*x; v[3] = 5 + 15*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (3*n - 6)*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.