Real-rooted atlas

A133214 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 10\,x + 4\,x^{2} + 22\,n + 13\,n x + 2\,n x^{2} + 12\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(80 + 122\,x + 76\,x^{2} + 16\,x^{3} + 154\,n + 205\,n x + 88\,n x^{2} + 12\,n x^{3} + 88\,n^{2} + 100\,n^{2} x + 24\,n^{2} x^{2} + 14\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(-26 - 96\,x - 110\,x^{2} - 48\,x^{3} - 8\,x^{4} - 22\,n - 74\,n x - 68\,n x^{2} - 16\,n x^{3} - 4\,n^{2} - 12\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 29 + 24\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02924457
1177234848503
2108920161200256164577
36693160501368049607203242135
4410051211761357807168018000192064389625
525033788092212291728632403197605980849281283608295
615232016225792104245129196544457632012759041863681228825633421185
792398774304809884219840897550085652460821345408468787255296029952512309404135

All 15 rows

Technical details
Node id
n-1ab71236799bc8c6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1c3940d3e6e6c9d9

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:        29,         24,          4;
  1:       177,        234,         84,          8;
  2:      1089,       2016,       1200,        256,         16;
  3:      6693,      16050,      13680,       4960,        720,         32;
  4:     41005,     121176,     135780,      71680,      18000,       1920,         64;
  5:    250337,     880922,    1229172,     863240,     319760,      59808,       4928,       128;
  6:   1523201,    6225792,   10424512,    9196544,    4576320,    1275904,     186368,     12288,      256;
  7:   9239877,   43048098,   84219840,   89755008,   56524608,   21345408,    4687872,    552960,    29952,     512;
  8:  55897533,  292512600,  655375140,  820346880,  627735360,  301948416,   90451200,  16158720,  1578240,   71680,   1024;
  9: 337339665, 1959583626, 4950164340, 7125725640, 6434049600, 3779825280, 1450428672, 355576320, 52926720, 4364800, 168960, 2048;
%F Formula
The row polynomials satisfy (2*n^3+12*n^2+22*n+12+(4*n^2+13*n+10)*x+(2*n+4)*x^2)*P(n,x) = (14*n^3+88*n^2+154*n+80+(12*n^3+100*n^2+205*n+122)*x+(24*n^2+88*n+76)*x^2+(12*n+16)*x^3)*P(n-1,x) - (4*n^2+22*n+26+(12*n^2+74*n+96)*x+(8*n^2+68*n+110)*x^2+(16*n+48)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 29 + 24*x + 4*x^2.
%t Mathematica
P[0, x_] := 29 + 24*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^3 + 88*n^2 + 154*n + 80 + (12*n^3 + 100*n^2 + 205*n + 122)*x + (24*n^2 + 88*n + 76)*x^2 + (12*n + 16)*x^3)*P[n - 1, x] - (4*n^2 + 22*n + 26 + (12*n^2 + 74*n + 96)*x + (8*n^2 + 68*n + 110)*x^2 + (16*n + 48)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 12*n^2 + 22*n + 12 + (4*n^2 + 13*n + 10)*x + (2*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 29 + 24*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^3 + 88*n^2 + 154*n + 80 + (12*n^3 + 100*n^2 + 205*n + 122)*x + (24*n^2 + 88*n + 76)*x^2 + (12*n + 16)*x^3)*v[i - 1] - (4*n^2 + 22*n + 26 + (12*n^2 + 74*n + 96)*x + (8*n^2 + 68*n + 110)*x^2 + (16*n + 48)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 12*n^2 + 22*n + 12 + (4*n^2 + 13*n + 10)*x + (2*n + 4)*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

Same fingerprint ?

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023