Real-rooted atlas

A169803 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 + 210\,x + 124\,x^{2} + 18\,x^{3} + 4\,x^{4} + 100\,n + 335\,n x + 314\,n x^{2} + 105\,n x^{3} + 16\,n x^{4} + 125\,n^{2} x + 265\,n^{2} x^{2} + 157\,n^{2} x^{3} + 25\,n^{2} x^{4} + 75\,n^{3} x^{2} + 85\,n^{3} x^{3} + 19\,n^{3} x^{4} + 15\,n^{4} x^{3} + 7\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(100 + 210\,x + 124\,x^{2} + 18\,x^{3} + 4\,x^{4} + 300\,n + 1380\,n x + 2152\,n x^{2} + 1483\,n x^{3} + 480\,n x^{4} + 72\,n x^{5} + 550\,n^{2} x + 1660\,n^{2} x^{2} + 1721\,n^{2} x^{3} + 737\,n^{2} x^{4} + 120\,n^{2} x^{5} + 300\,n^{3} x^{2} + 595\,n^{3} x^{3} + 371\,n^{3} x^{4} + 74\,n^{3} x^{5} + 55\,n^{4} x^{3} + 69\,n^{4} x^{4} + 20\,n^{4} x^{5} + 3\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 900\,x - 2435\,x^{2} - 3306\,x^{3} - 2426\,x^{4} - 924\,x^{5} - 144\,x^{6} - 375\,n x - 1850\,n x^{2} - 3500\,n x^{3} - 3193\,n x^{4} - 1408\,n x^{5} - 240\,n x^{6} - 300\,n^{2} x^{2} - 1040\,n^{2} x^{3} - 1325\,n^{2} x^{4} - 733\,n^{2} x^{5} - 148\,n^{2} x^{6} - 75\,n^{3} x^{3} - 185\,n^{3} x^{4} - 150\,n^{3} x^{5} - 40\,n^{3} x^{6} - 5\,n^{4} x^{4} - 9\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1246
23811
3520934
48402775
513808116190
62115219864435
734284459224251026

All 15 rows

Technical details
Node id
n-6980ec73d35c07b7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9a99cf3d0e7eaf56

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,   4;
  2:  3,   8;
  3:  5,  20,    9;
  4:  8,  40,   27;
  5: 13,  80,   81,   16;
  6: 21, 152,  198,   64;
  7: 34, 284,  459,  224,  25;
  8: 55, 520,  999,  640, 125;
  9: 89, 940, 2097, 1680, 500, 36;
%F Formula
The row polynomials satisfy (100*n+100+(125*n^2+335*n+210)*x+(75*n^3+265*n^2+314*n+124)*x^2+(15*n^4+85*n^3+157*n^2+105*n+18)*x^3+(n^5+7*n^4+19*n^3+25*n^2+16*n+4)*x^4)*P(n,x) = (300*n+100+(550*n^2+1380*n+210)*x+(300*n^3+1660*n^2+2152*n+124)*x^2+(55*n^4+595*n^3+1721*n^2+1483*n+18)*x^3+(3*n^5+69*n^4+371*n^3+737*n^2+480*n+4)*x^4+(2*n^5+20*n^4+74*n^3+120*n^2+72*n)*x^5)*P(n-1,x) - (125+(375*n+900)*x+(300*n^2+1850*n+2435)*x^2+(75*n^3+1040*n^2+3500*n+3306)*x^3+(5*n^4+185*n^3+1325*n^2+3193*n+2426)*x^4+(9*n^4+150*n^3+733*n^2+1408*n+924)*x^5+(4*n^4+40*n^3+148*n^2+240*n+144)*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[((300*n + 100 + (550*n^2 + 1380*n + 210)*x + (300*n^3 + 1660*n^2 + 2152*n + 124)*x^2 + (55*n^4 + 595*n^3 + 1721*n^2 + 1483*n + 18)*x^3 + (3*n^5 + 69*n^4 + 371*n^3 + 737*n^2 + 480*n + 4)*x^4 + (2*n^5 + 20*n^4 + 74*n^3 + 120*n^2 + 72*n)*x^5)*P[n - 1, x] - (125 + (375*n + 900)*x + (300*n^2 + 1850*n + 2435)*x^2 + (75*n^3 + 1040*n^2 + 3500*n + 3306)*x^3 + (5*n^4 + 185*n^3 + 1325*n^2 + 3193*n + 2426)*x^4 + (9*n^4 + 150*n^3 + 733*n^2 + 1408*n + 924)*x^5 + (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^6)*D[P[n - 1, x], x])/(100*n + 100 + (125*n^2 + 335*n + 210)*x + (75*n^3 + 265*n^2 + 314*n + 124)*x^2 + (15*n^4 + 85*n^3 + 157*n^2 + 105*n + 18)*x^3 + (n^5 + 7*n^4 + 19*n^3 + 25*n^2 + 16*n + 4)*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] = ((300*n + 100 + (550*n^2 + 1380*n + 210)*x + (300*n^3 + 1660*n^2 + 2152*n + 124)*x^2 + (55*n^4 + 595*n^3 + 1721*n^2 + 1483*n + 18)*x^3 + (3*n^5 + 69*n^4 + 371*n^3 + 737*n^2 + 480*n + 4)*x^4 + (2*n^5 + 20*n^4 + 74*n^3 + 120*n^2 + 72*n)*x^5)*v[i - 1] - (125 + (375*n + 900)*x + (300*n^2 + 1850*n + 2435)*x^2 + (75*n^3 + 1040*n^2 + 3500*n + 3306)*x^3 + (5*n^4 + 185*n^3 + 1325*n^2 + 3193*n + 2426)*x^4 + (9*n^4 + 150*n^3 + 733*n^2 + 1408*n + 924)*x^5 + (4*n^4 + 40*n^3 + 148*n^2 + 240*n + 144)*x^6)*deriv(v[i - 1], x))/(100*n + 100 + (125*n^2 + 335*n + 210)*x + (75*n^3 + 265*n^2 + 314*n + 124)*x^2 + (15*n^4 + 85*n^3 + 157*n^2 + 105*n + 18)*x^3 + (n^5 + 7*n^4 + 19*n^3 + 25*n^2 + 16*n + 4)*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

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.