Real-rooted atlas

A116412 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(156 + 216\,x + 96\,x^{2} - 52\,n + 292\,n x + 240\,n x^{2} - 64\,n x^{3} - 32\,n x^{4} - 52\,n^{2} x + 248\,n^{2} x^{2} + 112\,n^{2} x^{3} - 32\,n^{2} x^{4} - 24\,n^{3} x^{2} + 48\,n^{3} x^{3}\right)\,P_{n}(x) = \left(832 + 2400\,x + 1968\,x^{2} + 32\,x^{3} - 448\,x^{4} - 128\,x^{5} - 416\,n + 412\,n x + 2680\,n x^{2} + 2032\,n x^{3} + 160\,n x^{4} - 128\,n x^{5} - 572\,n^{2} x - 392\,n^{2} x^{2} + 752\,n^{2} x^{3} + 416\,n^{2} x^{4} - 192\,n^{3} x^{2} - 192\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 728\,x + 876\,x^{2} - 112\,x^{3} - 816\,x^{4} - 448\,x^{5} - 64\,x^{6} + 338\,n x + 1352\,n x^{2} + 1592\,n x^{3} + 368\,n x^{4} - 256\,n x^{5} - 64\,n x^{6} + 156\,n^{2} x^{2} + 552\,n^{2} x^{3} + 528\,n^{2} x^{4} + 96\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
213321257
34316814432387
41427601044512802538
5469318860604736160019216245
61549127683106833664180004608448102105
751164956414713220473615160061056125441024632772

All 15 rows

Technical details
Node id
n-a59b31e2ece04346 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2ae4e9c0c8806563

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:     4,      4;
  2:    13,     32,      12;
  3:    43,    168,     144,      32;
  4:   142,    760,    1044,     512,      80;
  5:   469,   3188,    6060,    4736,    1600,     192;
  6:  1549,  12768,   31068,   33664,   18000,    4608,     448;
  7:  5116,  49564,  147132,  204736,  151600,   61056,   12544,   1024;
  8: 16897, 188120,  659460, 1123072, 1068800,  594816,  191296,  32768,  2304;
  9: 55807, 701976, 2838564, 5724864, 6677520, 4774464, 2116800, 565248, 82944, 5120;
%F Formula
The row polynomials satisfy -(52*n-156+(52*n^2-292*n-216)*x+(24*n^3-248*n^2-240*n-96)*x^2-(48*n^3+112*n^2-64*n)*x^3+(32*n^2+32*n)*x^4)*P(n,x) = -(416*n-832+(572*n^2-412*n-2400)*x+(192*n^3+392*n^2-2680*n-1968)*x^2+(192*n^3-752*n^2-2032*n-32)*x^3-(416*n^2+160*n-448)*x^4+(128*n+128)*x^5)*P(n-1,x) + (169+(338*n+728)*x+(156*n^2+1352*n+876)*x^2+(552*n^2+1592*n-112)*x^3+(528*n^2+368*n-816)*x^4+(96*n^2-256*n-448)*x^5-(64*n+64)*x^6)*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[(-(416*n - 832 + (572*n^2 - 412*n - 2400)*x + (192*n^3 + 392*n^2 - 2680*n - 1968)*x^2 + (192*n^3 - 752*n^2 - 2032*n - 32)*x^3 - (416*n^2 + 160*n - 448)*x^4 + (128*n + 128)*x^5)*P[n - 1, x] + (169 + (338*n + 728)*x + (156*n^2 + 1352*n + 876)*x^2 + (552*n^2 + 1592*n - 112)*x^3 + (528*n^2 + 368*n - 816)*x^4 + (96*n^2 - 256*n - 448)*x^5 - (64*n + 64)*x^6)*D[P[n - 1, x], x])/(-52*n + 156 - (52*n^2 - 292*n - 216)*x - (24*n^3 - 248*n^2 - 240*n - 96)*x^2 + (48*n^3 + 112*n^2 - 64*n)*x^3 - (32*n^2 + 32*n)*x^4)]];
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] = (-(416*n - 832 + (572*n^2 - 412*n - 2400)*x + (192*n^3 + 392*n^2 - 2680*n - 1968)*x^2 + (192*n^3 - 752*n^2 - 2032*n - 32)*x^3 - (416*n^2 + 160*n - 448)*x^4 + (128*n + 128)*x^5)*v[i - 1] + (169 + (338*n + 728)*x + (156*n^2 + 1352*n + 876)*x^2 + (552*n^2 + 1592*n - 112)*x^3 + (528*n^2 + 368*n - 816)*x^4 + (96*n^2 - 256*n - 448)*x^5 - (64*n + 64)*x^6)*deriv(v[i - 1], x))/(-52*n + 156 - (52*n^2 - 292*n - 216)*x - (24*n^3 - 248*n^2 - 240*n - 96)*x^2 + (48*n^3 + 112*n^2 - 64*n)*x^3 - (32*n^2 + 32*n)*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.