Real-rooted atlas

basis-map coefficients of A021012 → basis A008277 → (θ+1)²

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

Recurrence

\(P_{n}(x) = \left(-8\,x + 18\,x^{2} + 12\,n x - 27\,n x^{2} - 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 8\,x^{2} + 8\,x^{3} + 6\,n x - 12\,n x^{2} - 12\,n x^{3} - 2\,n^{2} x + 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 8\,x^{3} + 3\,n x^{2} - 12\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(1 + 4\,x - n - 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 2\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -1 - 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-1-4-5
32241844
4-6-168-324-96-594
52414405400384060011304
6-120-14880-97200-124800-45000-4320-286320
7720181440195048040320002520000544320352809264240
8-5040-2560320-43817760-137168640-132300000-48263040-6914880-322560-371352240
94032041126400109771200050125824007006608000384072192091276416092897280326592018007718400
10-362880-741726720-30471033600-198016358400-385786800000-298204623360-104552985600-17418240000-1322697600-36288000-1036551116160
11362880014849049600930819859200846236160000022383345600000234476073216001137760525440027590492160003394923840001995840000043908480069735531398400
12-39916800-326838758400-31084569331200-390546609868800-1376530848000000-1902094277529600-1227138391987200-406262333030400-72020884320000-6805814400000-318775564800-5748019200-5413135130726400
1347900160078460462080001127869142400000194093747527680008991459981504000016073724103557120013414716479324160058234699992268800139483646894592001883673792000000140898799641600538014597120080951270400479557194434841600
14-6227020800-204022109491200-44216393405184000-1035353796212736000-6238701912643200000-14220706109312102400-15051548177842176000-8334172903440384000-2590101466977024000-468740236124160000-49778717096908800-3019158580838400-95765352883200-1220496076800-48036639885820185600
15871782912005712967778918400186351521946670080059080309490442240000459357498177177600000132031154374110351360017460351767337583104001208655674723008512000474023910971939328000110390818906368000000156063820328672256001336589663625216000670357470182400001794129232896000196151155200005396735982206080512000
Technical details
Node id
n-1e68a1963f2d6543 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-915ca4b22aed877c

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): no
  • 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:      -1,         -4;
   3:       2,         24,           18;
   4:      -6,       -168,         -324,           -96;
   5:      24,       1440,         5400,          3840,           600;
   6:    -120,     -14880,       -97200,       -124800,        -45000,         -4320;
   7:     720,     181440,      1950480,       4032000,       2520000,        544320,         35280;
   8:   -5040,   -2560320,    -43817760,    -137168640,    -132300000,     -48263040,      -6914880,      -322560;
   9:   40320,   41126400,   1097712000,    5012582400,    7006608000,    3840721920,     912764160,     92897280,     3265920;
  10: -362880, -741726720, -30471033600, -198016358400, -385786800000, -298204623360, -104552985600, -17418240000, -1322697600, -36288000;
%F Formula
The row polynomials satisfy P(n,x) = -((4*n^2-12*n+8)*x-(9*n^2-27*n+18)*x^2)*P(n-2,x) - ((2*n^2-6*n+4)*x-(4*n^2-12*n+8)*x^2-(4*n^2-12*n+8)*x^3)*d/dx P(n-2,x) - ((n^2-3*n+2)*x^2-(4*n^2-12*n+8)*x^3)*d^2/dx^2 P(n-2,x) - (n-1+(4*n-4)*x)*P(n-1,x) - ((2*n-2)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 - 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -1 - 4*x;
P[n_, x_] := P[n, x] = Expand[-((4*n^2 - 12*n + 8)*x - (9*n^2 - 27*n + 18)*x^2)*P[n - 2, x] - ((2*n^2 - 6*n + 4)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], x] - ((n^2 - 3*n + 2)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], {x, 2}] - (n - 1 + (4*n - 4)*x)*P[n - 1, x] - ((2*n - 2)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 - 4*x; for(n = 3, N, my(i = n); v[i] = -((4*n^2 - 12*n + 8)*x - (9*n^2 - 27*n + 18)*x^2)*v[i - 2] - ((2*n^2 - 6*n + 4)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(v[i - 2], x) - ((n^2 - 3*n + 2)*x^2 - (4*n^2 - 12*n + 8)*x^3)*deriv(deriv(v[i - 2], x), x) - (n - 1 + (4*n - 4)*x)*v[i - 1] - ((2*n - 2)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x)); 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

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.