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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	1764644880384	43845233541120	484638819287040	3129081072353280	13013693657395200	36196659358497792	67485297109063680	81745583105324160	59237931000310920	20885167980878850	1531578985264449							283755240966796875
9	23219011584000	640566091579392	7957909887713280	58641297133731840	283964107316060160	944794159526891520	2189897891189116416	3499594692941445120	3709205833404083760	2389263217012540440	758131597705902255	50542106513726817						13892656597734375000
10	303147415240704	9194728587264000	126832086132719616	1050444105178152960	5805488416239452160	22489957299431964672	62356414528774840320	123885652130127157248	173229937300601533440	163205056669779685440	94614823393696601424	27292737517412481180	1667889514952984961					675734730838330078125
11	3932000297680896	130050241138262016	1972269281968128000	18136988316978905088	112660130280356904960	498110906113344995328	1608031946909385474048	3821557404692058071040	6643368095478068807424	8257293677995339760640	7001496931133548505376	3689978112354167455536	975715366247496202185	55040353993448503713				32683496165037597656250
12	50714779349680128	1816584137528573952	30041605702938525696	303729469423091712000	2094822150611063537664	10409796037904978018304	38354539770727564640256	106130108496019441287168	220694940120966353602560	341026228901207532114432	381486967923384696941568	294062871107609037225792	142064157325635447038136	34675423015872557339190	1816331681783800622529			1573151824905765380859375
13	650946661526274048	25103815778091663360	449604574038322053120	4956864940984856739840	37586521841107599360000	207387392910495290228736	858808173127160686510080	2712213883787163499560960	6566800463191202929643520	12138221706653149448140800	16880798330609772839664384	17166913556552311362370560	12130093433188872785563920	5409365990476118944913640	1226023885204065420207075	59938945498865420543457		75399584507317749023437500
14	8319791650276638720	343699837285872697344	6627407365416199127040	79130405030744681349120	654306172210001089658880	3969136706420962492416000	18250090576123585540128768	64778673630162977496760320	179006116329952790971023360	385252293840550571872419840	640898106111286290861834240	810278319869269096303890432	755344196488301699944304640	492668410209517294675211520	204010374497956485922457280	43156040759183102791289040	1977985201462558877934081	3600330160224422515869140625

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.