Real-rooted atlas

A099040 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,x - 32\,x^{2} + 72\,x^{4} + 36\,n x + 30\,n x^{2} - 150\,n x^{3} - 198\,n x^{4} + 78\,n^{2} x^{2} + 96\,n^{2} x^{3} - 153\,n^{2} x^{4} - 162\,n^{2} x^{5} + 54\,n^{3} x^{3} + 54\,n^{3} x^{4} - 54\,n^{3} x^{5} + 9\,n^{4} x^{4}\right)\,P_{n}(x) = \left(8 + 88\,x + 344\,x^{2} + 456\,x^{3} - 432\,x^{4} - 1728\,x^{5} - 1296\,x^{6} + 72\,n x + 588\,n x^{2} + 1524\,n x^{3} + 612\,n x^{4} - 2808\,n x^{5} - 2916\,n x^{6} + 156\,n^{2} x^{2} + 984\,n^{2} x^{3} + 1674\,n^{2} x^{4} - 270\,n^{2} x^{5} - 1944\,n^{2} x^{6} + 108\,n^{3} x^{3} + 504\,n^{3} x^{4} + 432\,n^{3} x^{5} - 324\,n^{3} x^{6} + 18\,n^{4} x^{4} + 54\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(8 + 88\,x + 260\,x^{2} - 60\,x^{3} - 1296\,x^{4} - 1836\,x^{5} - 972\,x^{6} + 72\,n x + 504\,n x^{2} + 900\,n x^{3} - 756\,n x^{4} - 3348\,n x^{5} - 2268\,n x^{6} + 156\,n^{2} x^{2} + 876\,n^{2} x^{3} + 1134\,n^{2} x^{4} - 810\,n^{2} x^{5} - 1620\,n^{2} x^{6} + 108\,n^{3} x^{3} + 468\,n^{3} x^{4} + 324\,n^{3} x^{5} - 324\,n^{3} x^{6} + 18\,n^{4} x^{4} + 54\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122426
26120324450
316480259234566544
444172813932380163240086120
51205856628562592004536002799361061568
632819104255960141350437584004758912228614412492352
789660672974592674611224105600464693764572288017915904141996032

All 15 rows

Technical details
Node id
n-d8af10c408a348fe · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1cf9cc5cd63bfe33

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:    2,     24;
  2:    6,    120,      324;
  3:   16,    480,     2592,      3456;
  4:   44,   1728,    13932,     38016,     32400;
  5:  120,   5856,    62856,    259200,    453600,     279936;
  6:  328,  19104,   255960,   1413504,   3758400,    4758912,    2286144;
  7:  896,  60672,   974592,   6746112,  24105600,   46469376,   45722880,   17915904;
  8: 2448, 188928,  3538080,  29445120, 132224400,  343201536,  514382400,  412065792,  136048896;
  9: 6688, 579456, 12394944, 120517632, 651952800, 2130033024, 4302523008, 5249359872, 3537271296, 1007769600;
%F Formula
The row polynomials satisfy (4+(36*n-4)*x+(78*n^2+30*n-32)*x^2+(54*n^3+96*n^2-150*n)*x^3+(9*n^4+54*n^3-153*n^2-198*n+72)*x^4-(54*n^3+162*n^2)*x^5)*P(n,x) = (8+(72*n+88)*x+(156*n^2+588*n+344)*x^2+(108*n^3+984*n^2+1524*n+456)*x^3+(18*n^4+504*n^3+1674*n^2+612*n-432)*x^4+(54*n^4+432*n^3-270*n^2-2808*n-1728)*x^5-(324*n^3+1944*n^2+2916*n+1296)*x^6)*P(n-2,x) + (8+(72*n+88)*x+(156*n^2+504*n+260)*x^2+(108*n^3+876*n^2+900*n-60)*x^3+(18*n^4+468*n^3+1134*n^2-756*n-1296)*x^4+(54*n^4+324*n^3-810*n^2-3348*n-1836)*x^5-(324*n^3+1620*n^2+2268*n+972)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (72*n + 88)*x + (156*n^2 + 588*n + 344)*x^2 + (108*n^3 + 984*n^2 + 1524*n + 456)*x^3 + (18*n^4 + 504*n^3 + 1674*n^2 + 612*n - 432)*x^4 + (54*n^4 + 432*n^3 - 270*n^2 - 2808*n - 1728)*x^5 - (324*n^3 + 1944*n^2 + 2916*n + 1296)*x^6)*P[n - 2, x] + (8 + (72*n + 88)*x + (156*n^2 + 504*n + 260)*x^2 + (108*n^3 + 876*n^2 + 900*n - 60)*x^3 + (18*n^4 + 468*n^3 + 1134*n^2 - 756*n - 1296)*x^4 + (54*n^4 + 324*n^3 - 810*n^2 - 3348*n - 1836)*x^5 - (324*n^3 + 1620*n^2 + 2268*n + 972)*x^6)*P[n - 1, x])/(4 + (36*n - 4)*x + (78*n^2 + 30*n - 32)*x^2 + (54*n^3 + 96*n^2 - 150*n)*x^3 + (9*n^4 + 54*n^3 - 153*n^2 - 198*n + 72)*x^4 - (54*n^3 + 162*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] = 1; v[2] = 2 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = ((8 + (72*n + 88)*x + (156*n^2 + 588*n + 344)*x^2 + (108*n^3 + 984*n^2 + 1524*n + 456)*x^3 + (18*n^4 + 504*n^3 + 1674*n^2 + 612*n - 432)*x^4 + (54*n^4 + 432*n^3 - 270*n^2 - 2808*n - 1728)*x^5 - (324*n^3 + 1944*n^2 + 2916*n + 1296)*x^6)*v[i - 2] + (8 + (72*n + 88)*x + (156*n^2 + 504*n + 260)*x^2 + (108*n^3 + 876*n^2 + 900*n - 60)*x^3 + (18*n^4 + 468*n^3 + 1134*n^2 - 756*n - 1296)*x^4 + (54*n^4 + 324*n^3 - 810*n^2 - 3348*n - 1836)*x^5 - (324*n^3 + 1620*n^2 + 2268*n + 972)*x^6)*v[i - 1])/(4 + (36*n - 4)*x + (78*n^2 + 30*n - 32)*x^2 + (54*n^3 + 96*n^2 - 150*n)*x^3 + (9*n^4 + 54*n^3 - 153*n^2 - 198*n + 72)*x^4 - (54*n^3 + 162*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. [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.