Real-rooted atlas

A225470 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
216361870
364288396162910
4256192050405400194414560
510241152048960972008877629160276640
64096645124032001270080210470417146085248806086080
71638434406429675521354752035090496510572163810628811022480152152000
865536176947220127744125411328465502464104640076813777983369586408322645395204260256000
9262144884736012828672010450944005248115712167541696003376401408041035118400269414605447142567040132067936000
10104857643253760778567680802897920052345294848224460149760637681144320117706089600013388193941768368140844802142770112004490309824000
11419430420761804845416448005780865024047433749299226252353290249957909519360257517213465604419828714470447528296345728284719728326407071141369600166141463488000
12167772169814671362564082892839473381376039784613806082757709106380813434010992230446149162808320011038551170549761781251522592256182490655977676810532755874534402545610893056006645658539520000
136710886445801799681408405340162578927583232313084642590722654683580989441612866991177728708523079930265622413593963464704501753556555130887676843562020505675384458181107712420959922679065609927882482918400285763317199360000
142684354562113929216075572969472016231012761600233548815532032237682893127680017601139004866560960813368266752003872340694245703681142072369087569920241443868001236992035258121287730432003334531212678150144180774320518374912041697106428257280013145112591170560000
Technical details
Node id
n-c76459ebedf09334 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e42e4399c1128dea

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:      4,       3;
  2:     16,      36,        18;
  3:     64,     288,       396,        162;
  4:    256,    1920,      5040,       5400,       1944;
  5:   1024,   11520,     48960,      97200,      88776,       29160;
  6:   4096,   64512,    403200,    1270080,    2104704,     1714608,      524880;
  7:  16384,  344064,   2967552,   13547520,   35090496,    51057216,    38106288,    11022480;
  8:  65536, 1769472,  20127744,  125411328,  465502464,  1046400768,  1377798336,   958640832,   264539520;
  9: 262144, 8847360, 128286720, 1045094400, 5248115712, 16754169600, 33764014080, 41035118400, 26941460544, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (4+3*n*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[(4 + 3*n*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] = (4 + 3*n*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.