A204420 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 + 2\,x + 2\,n - 6\,n x + 4\,n^{2} x\right)\,P_{n-1}(x)\)

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

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	3	6													9
3	15	90	120												225
4	105	1260	4620	5040											11025
5	945	18900	132300	378000	362880										893025
6	10395	311850	3534300	18711000	45571680	39916800									108056025
7	135135	5675670	94594500	794593800	3511347840	7628100480	6227020800								18261468225
8	2027025	113513400	2610808200	31783752000	219534915600	851804553600	1695306412800	1307674368000							4108830350625
9	34459425	2481078600	75259384200	1250463614400	12377274109200	74194174454400	260511047596800	483353808537600	355687428096000						1187451971330625
10	654729075	58925616750	2278457181000	49497518070000	662826764199600	5642717059980000	30324117567744000	98278500640320000	172065063653683200	121645100408832000					428670161650355625
11	13749310575	1512424163250	72596359836000	1996399895490000	34708319637591600	396884299223412000	3006747634114224000	14799977891899200000	44890336605661747200	74821873843132416000	51090942171709440000				189043541287806830625
12	316234143225	41742906905700	2435002902832500	82650955673286000	1808469698782546800	26690748618759825600	269978650991673624000	1861813794375478944000	8521266555231215385600	24435413420856889651200	39034959834574319616000	25852016738884976640000			100004033341249813400625
13	7905853580625	1233313158577500	85920816714232500	3527275633531650000	94802315873535270000	1752350534738523720000	22763915446047735960000	208532535454391148000000	1330119940506067751040000	5723645268660151472640000	15637122048378491781120000	24067276319216716185600000	15511210043330985984000000		62502520838281133375390625
14	213458046676875	38849364495191250	3185647888605682500	155552855438745765000	5035810023324670590000	113864690181161899980000	1844775792429732522360000	21610647100394071639920000	182303078282646415174080000	1088282620489083650991360000	4440201273905645611883520000	11627076138580552567357440000	17314030647260071383859200000	10888869450418352160768000000	45564337691106946230659765625

Technical details

Node id: n-ed1a169c7e7a6346 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-181e40cb14f29d52

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:        3,          6;
  3:       15,         90,         120;
  4:      105,       1260,        4620,          5040;
  5:      945,      18900,      132300,        378000,         362880;
  6:    10395,     311850,     3534300,      18711000,       45571680,       39916800;
  7:   135135,    5675670,    94594500,     794593800,     3511347840,     7628100480,      6227020800;
  8:  2027025,  113513400,  2610808200,   31783752000,   219534915600,   851804553600,   1695306412800,   1307674368000;
  9: 34459425, 2481078600, 75259384200, 1250463614400, 12377274109200, 74194174454400, 260511047596800, 483353808537600, 355687428096000;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1+(4*n^2-6*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 - 1 + (4*n^2 - 6*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 - 1 + (4*n^2 - 6*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 T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
  - 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.