Real-rooted atlas

A121574 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-52\,n + 84\,n x + 288\,n x^{2} + 80\,n x^{3} - 52\,n^{2} x + 136\,n^{2} x^{2} + 128\,n^{2} x^{3} - 24\,n^{3} x^{2} + 48\,n^{3} x^{3}\right)\,P_{n}(x) = \left(260 + 1296\,x + 2144\,x^{2} + 1344\,x^{3} + 320\,x^{4} - 416\,n - 836\,n x + 352\,n x^{2} + 1296\,n x^{3} + 512\,n x^{4} - 572\,n^{2} x - 976\,n^{2} x^{2} - 80\,n^{2} x^{3} + 192\,n^{2} x^{4} - 192\,n^{3} x^{2} - 192\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 1066\,x + 2488\,x^{2} + 2608\,x^{3} + 1168\,x^{4} + 160\,x^{5} + 338\,n x + 1612\,n x^{2} + 2616\,n x^{3} + 1616\,n x^{4} + 256\,n x^{5} + 156\,n^{2} x^{2} + 552\,n^{2} x^{3} + 528\,n^{2} x^{4} + 96\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
210281250
33314013232337
4109620912480802201
5360256851484256152019214044
6118910200259202940816480441644888061
739273936412121217532813512056640120961024544711

All 15 rows

Technical details
Node id
n-64ce89a9f53b16fa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-69e2a6fa97325043

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:     3,      4;
  2:    10,     28,      12;
  3:    33,    140,     132,      32;
  4:   109,    620,     912,     480,      80;
  5:   360,   2568,    5148,    4256,    1520,     192;
  6:  1189,  10200,   25920,   29408,   16480,    4416,     448;
  7:  3927,  39364,  121212,  175328,  135120,   56640,   12096,   1024;
  8: 12970, 148756,  538248,  947744,  933680,  538176,  179200,  31744,  2304;
  9: 42837, 553220, 2300316, 4777120, 5743840, 4236288, 1937600, 533504, 80640, 5120;
%F Formula
The row polynomials satisfy -(52*n+(52*n^2-84*n)*x+(24*n^3-136*n^2-288*n)*x^2-(48*n^3+128*n^2+80*n)*x^3)*P(n,x) = -(416*n-260+(572*n^2+836*n-1296)*x+(192*n^3+976*n^2-352*n-2144)*x^2+(192*n^3+80*n^2-1296*n-1344)*x^3-(192*n^2+512*n+320)*x^4)*P(n-1,x) + (169+(338*n+1066)*x+(156*n^2+1612*n+2488)*x^2+(552*n^2+2616*n+2608)*x^3+(528*n^2+1616*n+1168)*x^4+(96*n^2+256*n+160)*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[(-(416*n - 260 + (572*n^2 + 836*n - 1296)*x + (192*n^3 + 976*n^2 - 352*n - 2144)*x^2 + (192*n^3 + 80*n^2 - 1296*n - 1344)*x^3 - (192*n^2 + 512*n + 320)*x^4)*P[n - 1, x] + (169 + (338*n + 1066)*x + (156*n^2 + 1612*n + 2488)*x^2 + (552*n^2 + 2616*n + 2608)*x^3 + (528*n^2 + 1616*n + 1168)*x^4 + (96*n^2 + 256*n + 160)*x^5)*D[P[n - 1, x], x])/(-52*n - (52*n^2 - 84*n)*x - (24*n^3 - 136*n^2 - 288*n)*x^2 + (48*n^3 + 128*n^2 + 80*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] = (-(416*n - 260 + (572*n^2 + 836*n - 1296)*x + (192*n^3 + 976*n^2 - 352*n - 2144)*x^2 + (192*n^3 + 80*n^2 - 1296*n - 1344)*x^3 - (192*n^2 + 512*n + 320)*x^4)*v[i - 1] + (169 + (338*n + 1066)*x + (156*n^2 + 1612*n + 2488)*x^2 + (552*n^2 + 2616*n + 2608)*x^3 + (528*n^2 + 1616*n + 1168)*x^4 + (96*n^2 + 256*n + 160)*x^5)*deriv(v[i - 1], x))/(-52*n - (52*n^2 - 84*n)*x - (24*n^3 - 136*n^2 - 288*n)*x^2 + (48*n^3 + 128*n^2 + 80*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.