Real-rooted atlas

A086645 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-72 + 156\,x - 18\,x^{2} + 132\,n - 398\,n x + 69\,n x^{2} - 80\,n^{2} + 376\,n^{2} x - 104\,n^{2} x^{2} + 16\,n^{3} - 156\,n^{3} x + 77\,n^{3} x^{2} + 24\,n^{4} x - 28\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-288 - 240\,x + 40\,x^{2} + 4\,x^{3} + 528\,n - 8\,n x - 496\,n x^{2} - 30\,n x^{3} - 320\,n^{2} + 544\,n^{2} x + 726\,n^{2} x^{2} + 64\,n^{3} - 432\,n^{3} x - 324\,n^{3} x^{2} + 63\,n^{3} x^{3} + 96\,n^{4} x + 8\,n^{4} x^{2} - 52\,n^{4} x^{3} + 16\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-72\,x - 68\,x^{2} + 6\,x^{3} + 2\,x^{4} + 96\,n x - 16\,n x^{2} - 126\,n x^{3} - 14\,n x^{4} - 32\,n^{2} x + 112\,n^{2} x^{2} + 146\,n^{2} x^{3} + 2\,n^{2} x^{4} - 48\,n^{3} x^{2} - 32\,n^{3} x^{3} + 16\,n^{3} x^{4} - 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
3832949
43219216216402
512810241440512253129
6512512010080640012503623398
7204824576622085734421000259249169817
88192114688354816430080235200564484802641204290

All 15 rows

Technical details
Node id
n-e89331d74c739da9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-415d66366a6cb218

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,       4;
   3:      8,      32,       9;
   4:     32,     192,     162,       16;
   5:    128,    1024,    1440,      512,       25;
   6:    512,    5120,   10080,     6400,     1250,      36;
   7:   2048,   24576,   62208,    57344,    21000,    2592,      49;
   8:   8192,  114688,  354816,   430080,   235200,   56448,    4802,     64;
   9:  32768,  524288, 1916928,  2883584,  2112000,  774144,  131712,   8192,    81;
  10: 131072, 2359296, 9953280, 17891328, 16473600, 8211456, 2173248, 276480, 13122, 100;
%F Formula
The row polynomials satisfy (16*n^3-80*n^2+132*n-72+(24*n^4-156*n^3+376*n^2-398*n+156)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (64*n^3-320*n^2+528*n-288+(96*n^4-432*n^3+544*n^2-8*n-240)*x+(16*n^5+8*n^4-324*n^3+726*n^2-496*n+40)*x^2+(12*n^5-52*n^4+63*n^3-30*n+4)*x^3)*P(n-1,x) - ((32*n^2-96*n+72)*x+(48*n^3-112*n^2+16*n+68)*x^2+(8*n^4+32*n^3-146*n^2+126*n-6)*x^3+(8*n^4-16*n^3-2*n^2+14*n-2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((64*n^3 - 320*n^2 + 528*n - 288 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*P[n - 1, x] - ((32*n^2 - 96*n + 72)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 80*n^2 + 132*n - 72 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((64*n^3 - 320*n^2 + 528*n - 288 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*v[i - 1] - ((32*n^2 - 96*n + 72)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 80*n^2 + 132*n - 72 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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