A008955 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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	1														4
2	27	21	4													52
3	513	642	265	36												1456
4	16929	29394	19017	5428	576											71344
5	863379	1922319	1704717	752253	165076	14400										5422144
6	63026667	171410931	193647825	116284281	39131656	6993936	518400									591013696
7	6239640033	20057988852	27570270294	21000887244	9571963713	2609850808	394024464	25401600								87470027008
8	804913564257	2986817524020	4840276154454	4473611753292	2578840102593	949276431864	217859607568	28494372096	1625702400							16881715212544
9	131200910973891	552049255120077	1030897232621622	1121261084297370	782713488739311	363618106703865	112402507014568	22291210864656	2573033630976	131681894400						4119138511860736
10	26371383105752091	124081991376524577	262415269268953722	328463201205933570	269451519666338511	151358588321407965	58954714580314668	15720784085252656	2746300846291776	283771423872000	13168189440000					1239860692070081536
11	6408246094697758113	33342861260291475222	78780831388915228263	111568805474585257872	105220766624838220143	69383770841729095326	32640384829906828089	10953670996934470236	2569565979964472944	401258858402200896	37536212322432000	1593350922240000				451309291913509679104
12	1851983121367652094657	10558874341860713507430	27569032292878473399975	43587824502158932394880	46474709542918522754895	35203700167236412249806	19424334217052063044665	7865826333620645143020	2319933191768296394800	485981311193120162880	68629240971099777024	5865692990957568000	229442532802560000			195416923398549691052032
13	627822278143634060088723	3892443549401915083015803	11130351711060263065347195	19435438963728340086460095	23121268875914238788644125	19788280269446374098261489	12534274627843603042358649	5949227609779197358032165	2115782002391341507007580	556816373903309825937520	105396154280840131937856	13586811648050477869056	1069083134091896832000	38775788043632640000		99271797086463243054432256
14	246734155310448185614868139	1652783481431104903402600287	5137147158129458740952545023	9819676448113049214786867555	12896004705125050500883319745	12308562845571615823191013677	8804472861554025318906200901	4794764277700570758008936049	1997550938456519894428283280	633522107412703696966931040	150556697917418897735331328	25997263216728503662358784	3083164754716009117310976	224779178983159406592000	7600054456551997440000	58471088483926850159060598784

Technical details

Node id: n-43f67bf0103635e1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-0f7f9997c3a05a90

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,               1;
  2:              27,              21,                4;
  3:             513,             642,              265,               36;
  4:           16929,           29394,            19017,             5428,             576;
  5:          863379,         1922319,          1704717,           752253,          165076,           14400;
  6:        63026667,       171410931,        193647825,        116284281,        39131656,         6993936,          518400;
  7:      6239640033,     20057988852,      27570270294,      21000887244,      9571963713,      2609850808,       394024464,       25401600;
  8:    804913564257,   2986817524020,    4840276154454,    4473611753292,   2578840102593,    949276431864,    217859607568,    28494372096,    1625702400;
  9: 131200910973891, 552049255120077, 1030897232621622, 1121261084297370, 782713488739311, 363618106703865, 112402507014568, 22291210864656, 2573033630976, 131681894400;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n^2 + 1 + n^2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n^2 + 1 + n^2*x)*v[i - 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    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 central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?

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.