A156319 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	3	8	4													15
3	3	14	20	8												45
4	3	20	48	48	16											135
5	3	26	88	144	112	32										405
6	3	32	140	320	400	256	64									1215
7	3	38	204	600	1040	1056	576	128								3645
8	3	44	280	1008	2240	3136	2688	1280	256							10935
9	3	50	368	1568	4256	7616	8960	6656	2816	512						32805
10	3	56	468	2304	7392	16128	24192	24576	16128	6144	1024					98415
11	3	62	580	3240	12000	30912	56448	72960	65280	38400	13312	2048				295245
12	3	68	704	4400	18480	54912	118272	185856	211200	168960	90112	28672	4096			885735
13	3	74	840	5808	27280	91872	228096	422400	582912	591360	428032	208896	61440	8192		2657205
14	3	80	988	7488	38896	146432	411840	878592	1427712	1757184	1610752	1064960	479232	131072	16384	7971615

Technical details

Node id: n-bd7d61845cd3400d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-a9636442fdb13e67

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 3,  2;
  2: 3,  8,   4;
  3: 3, 14,  20,    8;
  4: 3, 20,  48,   48,   16;
  5: 3, 26,  88,  144,  112,   32;
  6: 3, 32, 140,  320,  400,  256,   64;
  7: 3, 38, 204,  600, 1040, 1056,  576,  128;
  8: 3, 44, 280, 1008, 2240, 3136, 2688, 1280,  256;
  9: 3, 50, 368, 1568, 4256, 7616, 8960, 6656, 2816, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];
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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (2*n - 2)*x)*v[i - 1])/(n - 1)); 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A093562 → basis A013611

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.