Real-rooted atlas

A099040 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 16\,n x + 12\,n x^{2} + 6\,n^{2} x + 15\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(16 + 100\,x + 204\,x^{2} + 144\,x^{3} - 8\,n + 10\,n x + 144\,n x^{2} + 180\,n x^{3} - 18\,n^{2} x - 18\,n^{2} x^{2} + 36\,n^{2} x^{3} - 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(4 + 36\,x + 116\,x^{2} + 156\,x^{3} + 72\,x^{4} + 12\,n x + 78\,n x^{2} + 156\,n x^{3} + 90\,n x^{4} + 6\,n^{2} x^{2} + 24\,n^{2} x^{3} + 18\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121214
2660108174
3162408648641984
44486446449504648021536
5120292820952648009072046656226176
63289552853203533767516807931523265922320000
7896303363248641686528482112077448966531840223948823379968
8244894464117936073612802644488057200256734832005150822415116544232390656
966882897284131648301294081303905603550055046146461446561699843930301441007769602284576768
1018272878016140797441174694405948510401953626688426169900861339576325593121280292253184066512793622257360896
11499202635008469324804408888322557474560983900390425913115648470180505605780566425645954293760212840939524353564672215215767552
12136384784435215364166416048143361049233824046282005504142898043456313217031168485573626368520412221440366485492736152374763520282981703682067800129536
133726082319436849546252856959994884144239936020620067097673046938406418780323973123518345382912475526163456045135581736962851584860160107533047398418284971622419759290122240
14101798468185344157755686419794461184158625578880878895463680351411359040010371304074240227987716608003724414110720044579534782464379500232458242173299484262474968383651841175462461440187922146394112
Technical details
Node id
n-99755a57c875aa9f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6346f3bf74b3371e

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    2,     12;
  2:    6,     60,     108;
  3:   16,    240,     864,      864;
  4:   44,    864,    4644,     9504,      6480;
  5:  120,   2928,   20952,    64800,     90720,     46656;
  6:  328,   9552,   85320,   353376,    751680,    793152,    326592;
  7:  896,  30336,  324864,  1686528,   4821120,   7744896,   6531840,   2239488;
  8: 2448,  94464, 1179360,  7361280,  26444880,  57200256,  73483200,  51508224,  15116544;
  9: 6688, 289728, 4131648, 30129408, 130390560, 355005504, 614646144, 656169984, 393030144, 100776960;
%F Formula
The row polynomials satisfy (4*n+(6*n^2+16*n)*x+(3*n^3+15*n^2+12*n)*x^2)*P(n,x) = -(8*n-16+(18*n^2-10*n-100)*x+(6*n^3+18*n^2-144*n-204)*x^2-(36*n^2+180*n+144)*x^3)*P(n-1,x) + (4+(12*n+36)*x+(6*n^2+78*n+116)*x^2+(24*n^2+156*n+156)*x^3+(18*n^2+90*n+72)*x^4)*d/dx 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[Together[(-(8*n - 16 + (18*n^2 - 10*n - 100)*x + (6*n^3 + 18*n^2 - 144*n - 204)*x^2 - (36*n^2 + 180*n + 144)*x^3)*P[n - 1, x] + (4 + (12*n + 36)*x + (6*n^2 + 78*n + 116)*x^2 + (24*n^2 + 156*n + 156)*x^3 + (18*n^2 + 90*n + 72)*x^4)*D[P[n - 1, x], x])/(4*n + (6*n^2 + 16*n)*x + (3*n^3 + 15*n^2 + 12*n)*x^2)]];
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] = (-(8*n - 16 + (18*n^2 - 10*n - 100)*x + (6*n^3 + 18*n^2 - 144*n - 204)*x^2 - (36*n^2 + 180*n + 144)*x^3)*v[i - 1] + (4 + (12*n + 36)*x + (6*n^2 + 78*n + 116)*x^2 + (24*n^2 + 156*n + 156)*x^3 + (18*n^2 + 90*n + 72)*x^4)*deriv(v[i - 1], x))/(4*n + (6*n^2 + 16*n)*x + (3*n^3 + 15*n^2 + 12*n)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.