A008955 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	4	5	1													10
3	36	49	14	1												100
4	576	820	273	30	1											1700
5	14400	21076	7645	1023	55	1										44200
6	518400	773136	296296	44473	3003	91	1									1635400
7	25401600	38402064	15291640	2475473	191620	7462	140	1								81770000
8	1625702400	2483133696	1017067024	173721912	14739153	669188	16422	204	1							5315050000
9	131681894400	202759531776	84865562640	15088541896	1367593305	68943381	1999370	32946	285	1						435834100000
10	13168189440000	20407635072000	8689315795776	1593719752240	151847872396	8261931405	268880381	5293970	61446	385	1					44019244100000
11	1593350922240000	2482492033152000	1071814846360896	201529405816816	19967312312156	1151541572401	40796457506	909450751	12728936	108031	506	1				5370347780200000
12	229442532802560000	359072203696128000	156823829909121024	30092049283982400	3076822378767280	185789298737900	7026231453265	171757365650	2742417535	28285400	180895	650	1			778700428129000000
13	38775788043632640000	60912644957448192000	26862299458337581056	5242380158902146624	550075031295652720	34475213865472380	1373222414339685	36053226248115	635225929065	7522650135	58856655	290745	819	1		132379072781930000000
14	7600054456551997440000	11977654199703478272000	5325923338791614078976	1054368810603158319360	113057086292850079744	7307216948928239200	303626807076050640	8439654758970225	160557508344855	2109665355525	19058554515	115842675	451269	1015	1	26078677338040210000000

Technical details

Node id: n-04f83371218a2837 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d0a612563fb7dda7

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:            1,            1;
  2:            4,            5,           1;
  3:           36,           49,          14,           1;
  4:          576,          820,         273,          30,          1;
  5:        14400,        21076,        7645,        1023,         55,        1;
  6:       518400,       773136,      296296,       44473,       3003,       91,       1;
  7:     25401600,     38402064,    15291640,     2475473,     191620,     7462,     140,     1;
  8:   1625702400,   2483133696,  1017067024,   173721912,   14739153,   669188,   16422,   204,   1;
  9: 131681894400, 202759531776, 84865562640, 15088541896, 1367593305, 68943381, 1999370, 32946, 285, 1;

%F Formula

The row polynomials satisfy P(n,x) = (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[(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] = (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 (−∞, 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 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

Related ?

Same fingerprint ?

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.