A008955 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	16	26	10													52
3	288	628	440	100												1456
4	9216	24992	24756	10680	1700											71344
5	460800	1489216	1887592	1177656	362680	44200										5422144
6	33177600	124273152	191007616	154632136	69686232	16601560	1635400									591013696
7	3251404800	13837648896	24932403968	24704330128	14560857536	5111264480	990347200	81770000								87470027008
8	416179814400	1982560370688	4090794886144	4782760514304	3469571222928	1600697593280	458996632800	74839128000	5315050000							16881715212544
9	67421129932800	355301524832256	825278721951744	1110252383981056	954256900287264	543817850391456	205614657162560	49761662625600	6997846596000	435834100000						4119138511860736
10	13484225986560000	77869839089664000	200941198398406656	305403627713337344	302986870839539456	205143517007304864	96048534322049056	30719412898538560	6425497244385600	793949326196000	44019244100000					1239860692070081536
11	3263182688747520000	20489576630059008000	58127890381353418752	98422504111233249280	110582065324195704320	86609129358191590720	48271254380827064960	19152019108736316352	5302738706763019520	976046400754475200	107514474868112000	5370347780200000				451309291913509679104
12	939796614359285760000	6374159559325384704000	19711821041188340760576	36774225289331421511680	46118897909497183989760	40977828727167555253760	26460445018615975362880	12515113388535983528576	4304231518314515492800	1049998475897926688000	172490896871415160000	17136259016573840000	778700428129000000			195416923398549691052032
13	317651255653438586880000	2314231355493058609152000	7746202637006974576754688	15780697724796038400245760	21839805792596389845524480	21690718754397154954030080	15909861299910684065792640	8728383978489878244348288	3582399529241423436424320	1086618842966966854320000	236801664045185861040000	35115508015742535120000	3176364777525154800000	132379072781930000000		99271797086463243054432256
14	124519292216147926056960000	969755988717006376402944000	3492415010738866580090781696	7712035427610421044517011456	11670001322482604384294010880	12805203492865173541548113920	10509737224181227679734640640	6555769195650437032745678976	3123792259225144001214946176	1131687293703611423869031040	306890164370205327828720000	60415206959072688391920000	8162890071891140100240000	677636457702972055600000	26078677338040210000000	58471088483926850159060598784

Technical details

Node id: n-e9f9c578a5078389 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-1258bebed215369f

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:              2,               2;
  2:             16,              26,              10;
  3:            288,             628,             440,              100;
  4:           9216,           24992,           24756,            10680,            1700;
  5:         460800,         1489216,         1887592,          1177656,          362680,           44200;
  6:       33177600,       124273152,       191007616,        154632136,        69686232,        16601560,         1635400;
  7:     3251404800,     13837648896,     24932403968,      24704330128,     14560857536,      5111264480,       990347200,       81770000;
  8:   416179814400,   1982560370688,   4090794886144,    4782760514304,   3469571222928,   1600697593280,    458996632800,    74839128000,    5315050000;
  9: 67421129932800, 355301524832256, 825278721951744, 1110252383981056, 954256900287264, 543817850391456, 205614657162560, 49761662625600, 6997846596000, 435834100000;

%F Formula

The row polynomials satisfy P(n,x) = (2*n^2+(n^2+1)*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 + (n^2 + 1)*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 + (n^2 + 1)*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 (−∞, 0]
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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 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)

Related ?

Same fingerprint ?

A008955 → basis A013609

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.