Real-rooted atlas

A207606 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 215\,x + 399\,x^{2} + 109\,x^{3} - 40\,x^{4} + 35\,n x^{2} - 185\,n x^{3} - 144\,n x^{4} - 35\,n^{2} x^{2} - 115\,n^{2} x^{3} - 77\,n^{2} x^{4} + 8\,n^{2} x^{5} - 2\,n^{3} x^{4} + 8\,n^{3} x^{5} + n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n}(x) = \left(-25 - 215\,x - 399\,x^{2} + 191\,x^{3} + 262\,x^{4} - 18\,x^{5} + 35\,n x^{2} + 415\,n x^{3} + 300\,n x^{4} - 48\,n x^{5} + 35\,n^{2} x^{2} + 115\,n^{2} x^{3} + 77\,n^{2} x^{4} - 44\,n^{2} x^{5} - 2\,n^{3} x^{4} - 16\,n^{3} x^{5} - n^{4} x^{4} - 2\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(75 + 745\,x + 1987\,x^{2} + 1313\,x^{3} + 86\,x^{4} - 90\,x^{5} - 900\,n x^{3} - 1086\,n x^{4} - 294\,n x^{5} - 105\,n^{2} x^{2} - 395\,n^{2} x^{3} - 404\,n^{2} x^{4} - 50\,n^{2} x^{5} + 18\,n^{2} x^{6} + 36\,n^{3} x^{5} + 12\,n^{3} x^{6} + 3\,n^{4} x^{4} + 7\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 13 + 44\,x + 18\,x^{2}\)
\(P_{1}(x) = 34 + 176\,x + 153\,x^{2} + 32\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
013441875
13417615332395
289620837368501964
3233204037532560725729383
461064321501213984612512609843521
5159719696557556608039500125282009128197293
641815904419656928337621562593204229813008162878150
710946174160666801113350410501505775841937953891242932003850345

All 15 rows

Technical details
Node id
n-2163b10148741ca2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b5f4002bcc8be210

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:    13,      44,      18;
  1:    34,     176,     153,       32;
  2:    89,     620,     837,      368,       50;
  3:   233,    2040,    3753,     2560,      725,       72;
  4:   610,    6432,   15012,    13984,     6125,     1260,      98;
  5:  1597,   19696,   55755,    66080,    39500,    12528,    2009,     128;
  6:  4181,   59044,  196569,   283376,   215625,    93204,   22981,    3008,    162;
  7: 10946,  174160,  666801,  1133504,  1050150,   577584,  193795,   38912,   4293,   200;
  8: 28657,  507220, 2195694,  4302560,  4705925,  3151764, 1344560,  366848,  61965,  5900,  242;
  9: 75025, 1462128, 7061526, 15677632, 19790375, 15654240, 8129786, 2817792, 645894, 94000, 7865, 288;
%F Formula
The row polynomials satisfy (25+215*x-(35*n^2-35*n-399)*x^2-(115*n^2+185*n-109)*x^3+(n^4-2*n^3-77*n^2-144*n-40)*x^4+(2*n^4+8*n^3+8*n^2)*x^5)*P(n,x) = -(25+215*x-(35*n^2+35*n-399)*x^2-(115*n^2+415*n+191)*x^3+(n^4+2*n^3-77*n^2-300*n-262)*x^4+(2*n^4+16*n^3+44*n^2+48*n+18)*x^5)*P(n-2,x) + (75+745*x-(105*n^2-1987)*x^2-(395*n^2+900*n-1313)*x^3+(3*n^4-404*n^2-1086*n+86)*x^4+(7*n^4+36*n^3-50*n^2-294*n-90)*x^5+(2*n^4+12*n^3+18*n^2)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 44*x + 18*x^2, P(1,x) = 34 + 176*x + 153*x^2 + 32*x^3.
%t Mathematica
P[0, x_] := 13 + 44*x + 18*x^2;
P[1, x_] := 34 + 176*x + 153*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(25 + 215*x - (35*n^2 + 35*n - 399)*x^2 - (115*n^2 + 415*n + 191)*x^3 + (n^4 + 2*n^3 - 77*n^2 - 300*n - 262)*x^4 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*P[n - 2, x] + (75 + 745*x - (105*n^2 - 1987)*x^2 - (395*n^2 + 900*n - 1313)*x^3 + (3*n^4 - 404*n^2 - 1086*n + 86)*x^4 + (7*n^4 + 36*n^3 - 50*n^2 - 294*n - 90)*x^5 + (2*n^4 + 12*n^3 + 18*n^2)*x^6)*P[n - 1, x])/(25 + 215*x - (35*n^2 - 35*n - 399)*x^2 - (115*n^2 + 185*n - 109)*x^3 + (n^4 - 2*n^3 - 77*n^2 - 144*n - 40)*x^4 + (2*n^4 + 8*n^3 + 8*n^2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 44*x + 18*x^2; v[2] = 34 + 176*x + 153*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + 215*x - (35*n^2 + 35*n - 399)*x^2 - (115*n^2 + 415*n + 191)*x^3 + (n^4 + 2*n^3 - 77*n^2 - 300*n - 262)*x^4 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*v[i - 2] + (75 + 745*x - (105*n^2 - 1987)*x^2 - (395*n^2 + 900*n - 1313)*x^3 + (3*n^4 - 404*n^2 - 1086*n + 86)*x^4 + (7*n^4 + 36*n^3 - 50*n^2 - 294*n - 90)*x^5 + (2*n^4 + 12*n^3 + 18*n^2)*x^6)*v[i - 1])/(25 + 215*x - (35*n^2 - 35*n - 399)*x^2 - (115*n^2 + 185*n - 109)*x^3 + (n^4 - 2*n^3 - 77*n^2 - 144*n - 40)*x^4 + (2*n^4 + 8*n^3 + 8*n^2)*x^5)); 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. [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