A013618 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(15 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(312 + 990\,x + 363\,x^{2} + 132\,n + 363\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 936 + 2970\,x + 1089\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	936	2970	1089								4995
1	15984	92664	147015	35937							291600
2	248832	2109888	6115824	6468660	1185921						16129125
3	3670272	41057280	174065760	336370320	266832225	39135393					861131250
4	52254720	726713856	4064670720	11488340160	16650330840	10566556110	1291467969				44840334375
5	725594112	12070840320	83935450368	312979645440	663451644240	769245284808	406812410235	42618442977			2291839312500
6	9889579008	191556845568	1593350922240	7386319632384	20656656599040	35030246815872	33846792531552	15342639471720	1406408618241		115463861015625
7	132864344064	2937204965376	28446191566848	157741741301760	548434232704512	1227005401982976	1733997217385664	1436071054552992	569595490387605	46411484401953	5750772883593750

All 15 rows

Technical details

Node id: n-bb6ea5502c25c8fc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b593a3310b5e9356

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:            936,            2970,             1089;
  1:          15984,           92664,           147015,             35937;
  2:         248832,         2109888,          6115824,           6468660,            1185921;
  3:        3670272,        41057280,        174065760,         336370320,          266832225,           39135393;
  4:       52254720,       726713856,       4064670720,       11488340160,        16650330840,        10566556110,          1291467969;
  5:      725594112,     12070840320,      83935450368,      312979645440,       663451644240,       769245284808,        406812410235,         42618442977;
  6:     9889579008,    191556845568,    1593350922240,     7386319632384,     20656656599040,     35030246815872,      33846792531552,      15342639471720,       1406408618241;
  7:   132864344064,   2937204965376,   28446191566848,   157741741301760,    548434232704512,   1227005401982976,    1733997217385664,    1436071054552992,     569595490387605,      46411484401953;
  8:  1764644880384,  43845233541120,  484638819287040,  3129081072353280,  13013693657395200,  36196659358497792,   67485297109063680,   81745583105324160,   59237931000310920,   20885167980878850,   1531578985264449;
  9: 23219011584000, 640566091579392, 7957909887713280, 58641297133731840, 283964107316060160, 944794159526891520, 2189897891189116416, 3499594692941445120, 3709205833404083760, 2389263217012540440, 758131597705902255, 50542106513726817;

%F Formula

The row polynomials satisfy (11*n+15+11*x)*P(n,x) = (132*n+312+(363*n+990)*x+363*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 936 + 2970*x + 1089*x^2.

%t Mathematica

P[0, x_] := 936 + 2970*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((132*n + 312 + (363*n + 990)*x + 363*x^2)*P[n - 1, x])/(11*n + 15 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 936 + 2970*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 312 + (363*n + 990)*x + 363*x^2)*v[i - 1])/(11*n + 15 + 11*x)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients in expansion of (1+11x)^n.?
    
    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
    
    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.