Real-rooted atlas

A172431 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(200\,n + 170\,n x + 18\,n x^{2} - 28\,n x^{3} + 500\,n^{2} x + 160\,n^{2} x^{2} - 32\,n^{2} x^{3} + 600\,n^{3} x^{2} + 200\,n^{3} x^{3} + 8\,n^{3} x^{4} + 240\,n^{4} x^{3} + 32\,n^{4} x^{4} + 32\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-200 - 620\,x - 668\,x^{2} - 284\,x^{3} - 44\,x^{4} - 8\,x^{5} + 800\,n + 1530\,n x + 162\,n x^{2} - 1204\,n x^{3} - 664\,n x^{4} - 40\,n x^{5} + 2700\,n^{2} x + 5200\,n^{2} x^{2} + 3052\,n^{2} x^{3} + 428\,n^{2} x^{4} - 40\,n^{2} x^{5} + 3000\,n^{3} x^{2} + 4680\,n^{3} x^{3} + 2112\,n^{3} x^{4} + 248\,n^{3} x^{5} + 1120\,n^{4} x^{3} + 1296\,n^{4} x^{4} + 320\,n^{4} x^{5} + 128\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 525\,x - 885\,x^{2} - 771\,x^{3} - 378\,x^{4} - 108\,x^{5} - 16\,x^{6} - 750\,n x - 2850\,n x^{2} - 4220\,n x^{3} - 2996\,n x^{4} - 988\,n x^{5} - 112\,n x^{6} - 1200\,n^{2} x^{2} - 3560\,n^{2} x^{3} - 3860\,n^{2} x^{4} - 1804\,n^{2} x^{5} - 304\,n^{2} x^{6} - 600\,n^{3} x^{3} - 1400\,n^{3} x^{4} - 1056\,n^{3} x^{5} - 256\,n^{3} x^{6} - 80\,n^{4} x^{4} - 144\,n^{4} x^{5} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
28402775
32115219864435
4555209996401252339
5144168042664096162521612027
63775232165242118412775352834359963
798715880599949632078250332646860512292067

All 15 rows

Technical details
Node id
n-d044a3910145efa6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2d7099bd5283cb79

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,      8;
  2:    8,     40,     27;
  3:   21,    152,    198,      64;
  4:   55,    520,    999,     640,     125;
  5:  144,   1680,   4266,    4096,    1625,     216;
  6:  377,   5232,  16524,   21184,   12775,    3528,    343;
  7:  987,  15880,  59994,   96320,   78250,   33264,   6860,    512;
  8: 2584,  47288, 207981,  401408,  410875,  237168,  76146,  12288,   729;
  9: 6765, 138760, 696402, 1570560, 1941125, 1419480, 623280, 158208, 20655, 1000;
%F Formula
The row polynomials satisfy (200*n+(500*n^2+170*n)*x+(600*n^3+160*n^2+18*n)*x^2+(240*n^4+200*n^3-32*n^2-28*n)*x^3+(32*n^5+32*n^4+8*n^3)*x^4)*P(n,x) = (800*n-200+(2700*n^2+1530*n-620)*x+(3000*n^3+5200*n^2+162*n-668)*x^2+(1120*n^4+4680*n^3+3052*n^2-1204*n-284)*x^3+(128*n^5+1296*n^4+2112*n^3+428*n^2-664*n-44)*x^4+(96*n^5+320*n^4+248*n^3-40*n^2-40*n-8)*x^5)*P(n-1,x) - (125+(750*n+525)*x+(1200*n^2+2850*n+885)*x^2+(600*n^3+3560*n^2+4220*n+771)*x^3+(80*n^4+1400*n^3+3860*n^2+2996*n+378)*x^4+(144*n^4+1056*n^3+1804*n^2+988*n+108)*x^5+(64*n^4+256*n^3+304*n^2+112*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[((800*n - 200 + (2700*n^2 + 1530*n - 620)*x + (3000*n^3 + 5200*n^2 + 162*n - 668)*x^2 + (1120*n^4 + 4680*n^3 + 3052*n^2 - 1204*n - 284)*x^3 + (128*n^5 + 1296*n^4 + 2112*n^3 + 428*n^2 - 664*n - 44)*x^4 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^5)*P[n - 1, x] - (125 + (750*n + 525)*x + (1200*n^2 + 2850*n + 885)*x^2 + (600*n^3 + 3560*n^2 + 4220*n + 771)*x^3 + (80*n^4 + 1400*n^3 + 3860*n^2 + 2996*n + 378)*x^4 + (144*n^4 + 1056*n^3 + 1804*n^2 + 988*n + 108)*x^5 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^6)*D[P[n - 1, x], x])/(200*n + (500*n^2 + 170*n)*x + (600*n^3 + 160*n^2 + 18*n)*x^2 + (240*n^4 + 200*n^3 - 32*n^2 - 28*n)*x^3 + (32*n^5 + 32*n^4 + 8*n^3)*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] = ((800*n - 200 + (2700*n^2 + 1530*n - 620)*x + (3000*n^3 + 5200*n^2 + 162*n - 668)*x^2 + (1120*n^4 + 4680*n^3 + 3052*n^2 - 1204*n - 284)*x^3 + (128*n^5 + 1296*n^4 + 2112*n^3 + 428*n^2 - 664*n - 44)*x^4 + (96*n^5 + 320*n^4 + 248*n^3 - 40*n^2 - 40*n - 8)*x^5)*v[i - 1] - (125 + (750*n + 525)*x + (1200*n^2 + 2850*n + 885)*x^2 + (600*n^3 + 3560*n^2 + 4220*n + 771)*x^3 + (80*n^4 + 1400*n^3 + 3860*n^2 + 2996*n + 378)*x^4 + (144*n^4 + 1056*n^3 + 1804*n^2 + 988*n + 108)*x^5 + (64*n^4 + 256*n^3 + 304*n^2 + 112*n + 16)*x^6)*deriv(v[i - 1], x))/(200*n + (500*n^2 + 170*n)*x + (600*n^3 + 160*n^2 + 18*n)*x^2 + (240*n^4 + 200*n^3 - 32*n^2 - 28*n)*x^3 + (32*n^5 + 32*n^4 + 8*n^3)*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. [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