A028313 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

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

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

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	1														3
2	7	7	1													15
3	14	21	9	1												45
4	28	56	39	11	1											135
5	56	140	134	61	13	1										405
6	112	336	408	256	87	15	1									1215
7	224	784	1152	920	430	117	17	1								3645
8	448	1792	3088	2992	1780	664	151	19	1							10935
9	896	4032	7968	9072	6552	3108	966	189	21	1						32805
10	1792	8960	19968	26112	22176	12768	5040	1344	231	23	1					98415
11	3584	19712	48896	72192	70464	47712	22848	7728	1806	277	25	1				295245
12	7168	43008	117504	193280	213120	165888	93408	38304	11340	2360	327	27	1			885735
13	14336	93184	278016	504064	619520	544896	352704	170016	60984	16060	3014	381	29	1		2657205
14	28672	200704	649216	1286144	1743104	1709312	1250304	692736	291984	93104	22088	3776	439	31	1	7971615

Technical details

Node id: n-28029f075387bfc4 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-392ee40fda8b0784

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,    1;
  2:   7,    7,    1;
  3:  14,   21,    9,    1;
  4:  28,   56,   39,   11,    1;
  5:  56,  140,  134,   61,   13,    1;
  6: 112,  336,  408,  256,   87,   15,   1;
  7: 224,  784, 1152,  920,  430,  117,  17,   1;
  8: 448, 1792, 3088, 2992, 1780,  664, 151,  19,  1;
  9: 896, 4032, 7968, 9072, 6552, 3108, 966, 189, 21, 1;

%F Formula

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

%t Mathematica

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