Real-rooted atlas

A030528 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(20 + 32\,x + 20\,n + 52\,n x + 8\,n x^{2} - 16\,n x^{3} + 20\,n^{2} x + 16\,n^{2} x^{2} - 16\,n^{2} x^{3} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(120 + 272\,x + 176\,x^{2} - 64\,x^{3} - 64\,x^{4} - 40\,n + 156\,n x + 288\,n x^{2} + 48\,n x^{3} - 64\,n x^{4} - 60\,n^{2} x + 32\,n^{2} x^{2} + 80\,n^{2} x^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 90\,x + 72\,x^{2} - 64\,x^{3} - 112\,x^{4} - 32\,x^{5} + 50\,n x + 140\,n x^{2} + 48\,n x^{3} - 80\,n x^{4} - 32\,n x^{5} + 20\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
23161231
35407232149
488825225680684
51318070811528001923045
621352176439684400230444813257
73466840681171218000149766272102456754
8551240889231232616007104047040163842304239787
989226418684775041869602770562540161392644147251201002429
101444080380881822725204009446401113280843776393984102400112644154328
112337276758044111681356800291936042179204108288264729610752002478082457617091729
123771286414796089715233608008369280143472001713766414183424793600028497925898245324869885585
13610225802841841905024798840022610880449052806368051264634112464128002292224073728001384448114688284238558
149873939253847639545601835464058218240131539520216252416260658432229498880145339392641925121869004832112642457601150734519
Technical details
Node id
n-a2fcb766553882c3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b8bb14df92568efa

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,    4;
  2:  3,   16,    12;
  3:  5,   40,    72,    32;
  4:  8,   88,   252,   256,     80;
  5: 13,  180,   708,  1152,    800,    192;
  6: 21,  352,  1764,  3968,   4400,   2304,    448;
  7: 34,  668,  4068, 11712,  18000,  14976,   6272,   1024;
  8: 55, 1240,  8892, 31232,  61600,  71040,  47040,  16384,  2304;
  9: 89, 2264, 18684, 77504, 186960, 277056, 254016, 139264, 41472, 5120;
%F Formula
The row polynomials satisfy (20*n+20+(20*n^2+52*n+32)*x+(8*n^3+16*n^2+8*n)*x^2-(16*n^2+16*n)*x^3)*P(n,x) = -(40*n-120+(60*n^2-156*n-272)*x+(16*n^3-32*n^2-288*n-176)*x^2-(80*n^2+48*n-64)*x^3+(64*n+64)*x^4)*P(n-1,x) + (25+(50*n+90)*x+(20*n^2+140*n+72)*x^2+(48*n^2+48*n-64)*x^3+(16*n^2-80*n-112)*x^4-(32*n+32)*x^5)*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[(-(40*n - 120 + (60*n^2 - 156*n - 272)*x + (16*n^3 - 32*n^2 - 288*n - 176)*x^2 - (80*n^2 + 48*n - 64)*x^3 + (64*n + 64)*x^4)*P[n - 1, x] + (25 + (50*n + 90)*x + (20*n^2 + 140*n + 72)*x^2 + (48*n^2 + 48*n - 64)*x^3 + (16*n^2 - 80*n - 112)*x^4 - (32*n + 32)*x^5)*D[P[n - 1, x], x])/(20*n + 20 + (20*n^2 + 52*n + 32)*x + (8*n^3 + 16*n^2 + 8*n)*x^2 - (16*n^2 + 16*n)*x^3)]];
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] = (-(40*n - 120 + (60*n^2 - 156*n - 272)*x + (16*n^3 - 32*n^2 - 288*n - 176)*x^2 - (80*n^2 + 48*n - 64)*x^3 + (64*n + 64)*x^4)*v[i - 1] + (25 + (50*n + 90)*x + (20*n^2 + 140*n + 72)*x^2 + (48*n^2 + 48*n - 64)*x^3 + (16*n^2 - 80*n - 112)*x^4 - (32*n + 32)*x^5)*deriv(v[i - 1], x))/(20*n + 20 + (20*n^2 + 52*n + 32)*x + (8*n^3 + 16*n^2 + 8*n)*x^2 - (16*n^2 + 16*n)*x^3)); 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.