Real-rooted atlas

A225470 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(2 + 4\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
2405216108
3440732384641620
46160120088304243225629160
510472022877618920074560140801024612360
6209440049944004699104224800057984076800409614696640
7481712001232488001280569927050041622328320408576040140816384396809280
812524512003397153600382247699223452387848625379841955430402677964820316166553611904278400
93632108480010352725920012444044716883301832704343945566729120900096155878195216603545610027008262144392841187200
1011622747136003458156633600439620334617631634204352001433833144320429447029760863646228481154826240098500608048496640104857614142282739200
1140679614976000125684581030400167699743650560128304528616704628378417920002076597861888047405499187207496476753928066826240056374067202306867204194304551549026828800
121545825369088000493873253905920068753285828428805546371062036992290105610256281610404585546854402632048113868804744881133977660639846727685368945049603131572224010821304321677721623165059126809600
1363378840132608000208671335577779200301643402052794880254902527874888192141128784453223424542630251523543041495323148560384029982205104783364384186169425924626861339443234315226316801696302366725016387584671088641042427660706432000
1427886689658347520009435054125952716800141069950326340915201242228483470625996872292766274413834242952088244516483072874994285975986176191734628403462144312833011873873923789493457125376336061449371648211898209402888992420003842301834035226843545650036527713908736000
Technical details
Node id
n-2e1d794b12bde966 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d5983a4011f3413f

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:           5,            4;
  2:          40,           52,           16;
  3:         440,          732,          384,          64;
  4:        6160,        12008,         8304,        2432,         256;
  5:      104720,       228776,       189200,       74560,       14080,       1024;
  6:     2094400,      4994400,      4699104,     2248000,      579840,      76800,       4096;
  7:    48171200,    123248800,    128056992,    70500416,    22328320,    4085760,     401408,     16384;
  8:  1252451200,   3397153600,   3822476992,  2345238784,   862537984,  195543040,   26779648,   2031616,    65536;
  9: 36321084800, 103527259200, 124440447168, 83301832704, 34394556672, 9120900096, 1558781952, 166035456, 10027008, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+2+4*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[(3*n + 2 + 4*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] = (3*n + 2 + 4*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

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.