Real-rooted atlas

A249252 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-102 - 234\,x - 180\,x^{2} - 48\,x^{3} + 34\,n + 95\,n x + 70\,n x^{2} - 4\,n x^{3} - 8\,n x^{4} + 17\,n^{2} x + 54\,n^{2} x^{2} + 44\,n^{2} x^{3} + 8\,n^{2} x^{4} + 4\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-102 - 234\,x - 180\,x^{2} - 48\,x^{3} - 238\,n - 733\,n x - 690\,n x^{2} - 188\,n x^{3} + 8\,n x^{4} - 187\,n^{2} x - 338\,n^{2} x^{2} - 156\,n^{2} x^{3} - 8\,n^{2} x^{4} - 28\,n^{3} x^{2} - 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(289 + 1377\,x + 2546\,x^{2} + 2302\,x^{3} + 1044\,x^{4} + 216\,x^{5} + 16\,x^{6} + 289\,n x + 1037\,n x^{2} + 1364\,n x^{3} + 792\,n x^{4} + 192\,n x^{5} + 16\,n x^{6} + 68\,n^{2} x^{2} + 148\,n^{2} x^{3} + 96\,n^{2} x^{4} + 16\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15813
217482489
36124824064613
4217115216568961604081
5773506496487872288038426621
62753214565090455552308808448896170889
79805886002515683438722569601071362329620481083285

All 15 rows

Technical details
Node id
n-6705c85e93b23d1a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-01dea2aba2bfe03b

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:      5,       8;
  2:     17,      48,      24;
  3:     61,     248,     240,       64;
  4:    217,    1152,    1656,      896,      160;
  5:    773,    5064,    9648,     7872,     2880,     384;
  6:   2753,   21456,   50904,    55552,    30880,    8448,     896;
  7:   9805,   88600,  251568,   343872,   256960,  107136,   23296,    2048;
  8:  34921,  358944, 1186680,  1949312,  1831200, 1029120,  341376,   61440,   4608;
  9: 124373, 1432936, 5405808, 10371008, 11740480, 8313216, 3721984, 1021952, 156672, 10240;
%F Formula
The row polynomials satisfy (34*n-102+(17*n^2+95*n-234)*x+(4*n^3+54*n^2+70*n-180)*x^2+(8*n^3+44*n^2-4*n-48)*x^3+(8*n^2-8*n)*x^4)*P(n,x) = -(238*n+102+(187*n^2+733*n+234)*x+(28*n^3+338*n^2+690*n+180)*x^2+(24*n^3+156*n^2+188*n+48)*x^3+(8*n^2-8*n)*x^4)*P(n-1,x) + (289+(289*n+1377)*x+(68*n^2+1037*n+2546)*x^2+(148*n^2+1364*n+2302)*x^3+(96*n^2+792*n+1044)*x^4+(16*n^2+192*n+216)*x^5+(16*n+16)*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[(-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*P[n - 1, x] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*D[P[n - 1, x], x])/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*x^3 + (8*n^2 - 8*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] = (-(238*n + 102 + (187*n^2 + 733*n + 234)*x + (28*n^3 + 338*n^2 + 690*n + 180)*x^2 + (24*n^3 + 156*n^2 + 188*n + 48)*x^3 + (8*n^2 - 8*n)*x^4)*v[i - 1] + (289 + (289*n + 1377)*x + (68*n^2 + 1037*n + 2546)*x^2 + (148*n^2 + 1364*n + 2302)*x^3 + (96*n^2 + 792*n + 1044)*x^4 + (16*n^2 + 192*n + 216)*x^5 + (16*n + 16)*x^6)*deriv(v[i - 1], x))/(34*n - 102 + (17*n^2 + 95*n - 234)*x + (4*n^3 + 54*n^2 + 70*n - 180)*x^2 + (8*n^3 + 44*n^2 - 4*n - 48)*x^3 + (8*n^2 - 8*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.