A128813 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 12\,n x - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - 3\,n^{3} - 6\,n^{3} x + 9\,n^{3} x^{2} + n^{4} + 6\,n^{4} x + 9\,n^{4} x^{2} - n^{5} - 6\,n^{5} x - 9\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(4\,n - 12\,n x + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	8	60	81													149
3	56	708	2187	1944												4895
4	616	11148	55917	108864	72900											249445
5	9856	233808	1647162	5096844	7290000	3936600										18214270
6	216832	6385632	58332420	250492176	561756465	637729200	289340100									1804252825
7	6288128	221611104	2496229812	13797504144	42592615965	75119198472	70888324500	27776649600								232898421725
8	232660736	9557846496	128261501892	869964746256	3438589850145	8299413372528	12087887013972	9777380659200	3374862926400							37986150577625
9	10702393856	502479337536	7835493002472	63105448668336	304981684037370	938824570859778	1863200408815872	2314747792478880	1640183382230400	506229438960000						7639621400784500
10	599334055936	31670632874496	563151244178592	5257713585970656	30094473093937020	112960549407546828	285062952784196097	480972239184096000	521525328383794500	329049135324000000	91880643171240000					1857398456145890125
11	40155381747712	2359268688741888	47137311323690976	500938738723182240	3317613809821517700	14718803617425073428	45193104749684455767	96730813912501091664	142078763279971615500	136781864311142790000	77822904766040280000	19846218924987840000				537040562845692026875
12	3172275158069248	205174945068538368	4551950904319989792	54281001492122981472	408616072052430703500	2094371443574462970972	7588488662782117051437	19727661740717486412288	36688609061583670005048	47746245733372900440000	41355661350205336266000	21433916438986867200000	5031016497484417440000			182133628300376358068125
13	291849314542370816	20608157182611339264	502798623203005522560	6650762266447790579712	56116070388010592149320	326544798013226891796024	1365198261754421224986786	4182553342934029269978840	9434210145783555654838368	15521532689484020075344560	18142918437950772938604000	14288388183593399201256000	6801934304598932378880000	1479118850260418727360000		71605689545656934905006250
14	30936027341491306496	2368329729518495576064	63034008328302443193600	916156221988728120924672	8567041103542940308589520	55825623196070054362821504	264716229015829532583638136	934822028582598743613000000	2482218616355303521961618433	4947256016109550607179952160	7301360431328994887598914040	7749135665229438652417056000	5596917503938734309589890000	2464212004533857599781760000	499202611962891320484000000	32305215360852886060432240625

Technical details

Node id: n-9fc6fba32e88da55 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f000450fe779fa64

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,            6;
  2:           8,           60,            81;
  3:          56,          708,          2187,           1944;
  4:         616,        11148,         55917,         108864,           72900;
  5:        9856,       233808,       1647162,        5096844,         7290000,         3936600;
  6:      216832,      6385632,      58332420,      250492176,       561756465,       637729200,        289340100;
  7:     6288128,    221611104,    2496229812,    13797504144,     42592615965,     75119198472,      70888324500,      27776649600;
  8:   232660736,   9557846496,  128261501892,   869964746256,   3438589850145,   8299413372528,   12087887013972,    9777380659200,    3374862926400;
  9: 10702393856, 502479337536, 7835493002472, 63105448668336, 304981684037370, 938824570859778, 1863200408815872, 2314747792478880, 1640183382230400, 506229438960000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(6*n^5-6*n^4+6*n^3+6*n^2-12*n)*x+(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(12*n^3-12*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (12*n^3 - 12*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] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (12*n^3 - 12*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 basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.