Real-rooted atlas

A094638 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
2455718120
358511467471622640
49945265022645111718194476560
5208845705717953001642843216594291602756160
65221125214021353652793133225093169860244627692524880118514880
7151412625730305540150875483417306142481190321649490909572112403052110224805925744000
849966166252773398582067316242482933203862008081535636944767735452154911304443061415088264539520337767408000
918487481512511610661242153239518588974527039283641455098186130533838420833687178190068363253153288024492446169616714256704021617114112000
10757986742012553149955546565167652245874384313271663962194384014348227593322669812572757188210553039522752498685989641973627936157630662303371202142770112001534815101952000
113410940339056252641883624459550929829959738392519629748992153402276186105809940872719261505728463419117578701679471959719233725760633713782350367327892775841024081108280666289607071141369600119715577952256000
12167136076613756251417316828191204505507244907523561251296596486121337998205998288218623273523267081187227341981915695498444540903115849525778608365051069645765807368416004421808234034433844568292505728433629632575296025456108930560010175824125941760000
13885821206052908125816360988820703322534715933639820572175901978689037728027691597463557146025168772036548626975679891591922734780477473015811361123731019995286677224822730953516760228399505407028911908803548662471407700015497618686726302721826072944375622409927882482918400936175819586641920000
1450491808745015763125502530254282023035075232167983277446800942565993477403875137891831289385276969080117828718317674113774681090897195130924820194517347951583389134481935980042198348686269274847573284653197355201125018040165080196470391668913642582686475233115950475498614267191519104823539566790396288041697106428257280092681406139077550080000
Technical details
Node id
n-e2a9d8ea2cc2630f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b7f27fcb8eaa9029

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,             3;
  2:           45,            57,            18;
  3:          585,          1146,           747,           162;
  4:         9945,         26502,         26451,         11718,          1944;
  5:       208845,        705717,        953001,        642843,        216594,         29160;
  6:      5221125,      21402135,      36527931,      33225093,      16986024,       4627692,        524880;
  7:    151412625,     730305540,    1508754834,    1730614248,    1190321649,     490909572,     112403052,     11022480;
  8:   4996616625,   27733985820,   67316242482,   93320386200,   80815356369,   44767735452,   15491130444,   3061415088,   264539520;
  9: 184874815125, 1161066124215, 3239518588974, 5270392836414, 5509818613053, 3838420833687, 1781900683632, 531532880244, 92446169616, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+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*n + 1 + 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*n + 1 + 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

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.