Real-rooted atlas

A121314 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(12 - 12\,x - 32\,n + 74\,n x + 16\,n^{2} - 104\,n^{2} x + 12\,n^{2} x^{2} + 40\,n^{3} x - 32\,n^{3} x^{2} + 16\,n^{4} x^{2}\right)\,P_{n}(x) = \left(8 + 58\,x + 128\,x^{2} + 36\,x^{3} - 24\,n - 138\,n x - 132\,n x^{2} - 48\,n x^{3} + 16\,n^{2} + 24\,n^{2} x - 164\,n^{2} x^{2} - 100\,n^{2} x^{3} + 40\,n^{3} x + 144\,n^{3} x^{2} + 48\,n^{3} x^{3} + 16\,n^{4} x^{2} + 48\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 29\,x^{2} + 58\,x^{3} + 24\,x^{4} - 8\,n x - 40\,n x^{2} - 36\,n x^{3} - 16\,n x^{4} - 20\,n^{2} x^{2} - 100\,n^{2} x^{3} - 80\,n^{2} x^{4} - 8\,n^{3} x^{3} - 32\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1044
204913
304271647
404459625170
5046324025036593
60481448875540491997
70499720210025201029646536

All 15 rows

Technical details
Node id
n-359a01e1c9d099c3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f8028a91b50975db

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: 0, 4;
  2: 0, 4,   9;
  3: 0, 4,  27,   16;
  4: 0, 4,  45,   96,   25;
  5: 0, 4,  63,  240,  250,    36;
  6: 0, 4,  81,  448,  875,   540,    49;
  7: 0, 4,  99,  720, 2100,  2520,  1029,    64;
  8: 0, 4, 117, 1056, 4125,  7560,  6174,  1792,   81;
  9: 0, 4, 135, 1456, 7150, 17820, 22638, 13440, 2916, 100;
%F Formula
The row polynomials satisfy (16*n^2-32*n+12+(40*n^3-104*n^2+74*n-12)*x+(16*n^4-32*n^3+12*n^2)*x^2)*P(n,x) = (16*n^2-24*n+8+(40*n^3+24*n^2-138*n+58)*x+(16*n^4+144*n^3-164*n^2-132*n+128)*x^2+(48*n^4+48*n^3-100*n^2-48*n+36)*x^3)*P(n-1,x) - ((8*n-4)*x+(20*n^2+40*n-29)*x^2+(8*n^3+100*n^2+36*n-58)*x^3+(32*n^3+80*n^2+16*n-24)*x^4)*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[((16*n^2 - 24*n + 8 + (40*n^3 + 24*n^2 - 138*n + 58)*x + (16*n^4 + 144*n^3 - 164*n^2 - 132*n + 128)*x^2 + (48*n^4 + 48*n^3 - 100*n^2 - 48*n + 36)*x^3)*P[n - 1, x] - ((8*n - 4)*x + (20*n^2 + 40*n - 29)*x^2 + (8*n^3 + 100*n^2 + 36*n - 58)*x^3 + (32*n^3 + 80*n^2 + 16*n - 24)*x^4)*D[P[n - 1, x], x])/(16*n^2 - 32*n + 12 + (40*n^3 - 104*n^2 + 74*n - 12)*x + (16*n^4 - 32*n^3 + 12*n^2)*x^2)]];
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] = ((16*n^2 - 24*n + 8 + (40*n^3 + 24*n^2 - 138*n + 58)*x + (16*n^4 + 144*n^3 - 164*n^2 - 132*n + 128)*x^2 + (48*n^4 + 48*n^3 - 100*n^2 - 48*n + 36)*x^3)*v[i - 1] - ((8*n - 4)*x + (20*n^2 + 40*n - 29)*x^2 + (8*n^3 + 100*n^2 + 36*n - 58)*x^3 + (32*n^3 + 80*n^2 + 16*n - 24)*x^4)*deriv(v[i - 1], x))/(16*n^2 - 32*n + 12 + (40*n^3 - 104*n^2 + 74*n - 12)*x + (16*n^4 - 32*n^3 + 12*n^2)*x^2)); 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.