A269944 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-544f9159ada5f1ab · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d0a612563fb7dda7

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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;
  2: 1,   1;
  3: 1,   5,     4;
  4: 1,  14,    49,     36;
  5: 1,  30,   273,    820,      576;
  6: 1,  55,  1023,   7645,    21076,     14400;
  7: 1,  91,  3003,  44473,   296296,    773136,     518400;
  8: 1, 140,  7462, 191620,  2475473,  15291640,   38402064,   25401600;
  9: 1, 204, 16422, 669188, 14739153, 173721912, 1017067024, 2483133696, 1625702400;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n^2-2*n+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[(1 + (n^2 - 2*n + 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] = (1 + (n^2 - 2*n + 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 Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
  - 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
  - Parent-row provers of polytool 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.