A128813 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 - 4\,n + 4\,n x - n^{2} x^{2} - 4\,n^{3} x + n^{3} x^{2} + n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(8\,n - 4\,n x + 4\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	8	9													18
3	1	20	81	72												174
4	1	40	381	1152	900											2474
5	1	70	1281	8772	22500	16200										48824
6	1	112	3486	44640	252765	583200	396900									1281104
7	1	168	8190	174784	1815165	9076104	19448100	12700800								43223312
8	1	240	17262	567904	9680445	87491232	400644468	812851200	514382400							1825635152
9	1	330	33462	1603624	41625045	610235262	4993934148	21417423840	41664974400	25719120000						94448950112
10	1	440	60687	4057504	151874195	3357488232	44150696793	335321856000	1366868074500	2571912000000	1556006760000					5877772868352
11	1	572	104247	9397960	486618275	15385924476	302677290657	3665545842672	26264515882500	102808904130000	188276817960000	112032486720000				433366829871360
12	1	728	171171	20239648	1402919375	60933395016	1702796417973	30647064324096	347916163576968	2379066947280000	9009280792314000	16132678087680000	9466745127840000			37368099336158976
13	1	910	270543	41008396	3705179335	214132190766	8171891855505	207737891793288	3485409373756296	37557256820062320	247153882215042000	910508553106488000	1599879926604960000	927741022528320000		3726542178310927360
14	1	1120	413868	78884416	9087531310	680984786976	34403085224340	1188364914453888	28024447841838441	444188350424963520	4585017044932239960	29220862334102208000	105170227842467970000	181837240415550720000	104370865034436000000	425657648927907235840

Technical details

Node id: n-f8871b17e8bc3bb1 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-03bb4e5ed48317f0

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: 1,   2;
  2: 1,   8,     9;
  3: 1,  20,    81,      72;
  4: 1,  40,   381,    1152,      900;
  5: 1,  70,  1281,    8772,    22500,     16200;
  6: 1, 112,  3486,   44640,   252765,    583200,     396900;
  7: 1, 168,  8190,  174784,  1815165,   9076104,   19448100,    12700800;
  8: 1, 240, 17262,  567904,  9680445,  87491232,  400644468,   812851200,   514382400;
  9: 1, 330, 33462, 1603624, 41625045, 610235262, 4993934148, 21417423840, 41664974400, 25719120000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(4*n^3-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (8*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (8*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (8*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? 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.