Real-rooted atlas

A334218 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-12 + 16\,n - 7\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(24 + 36\,x + 12\,x^{2} - 74\,n - 114\,n x - 40\,n x^{2} + 85\,n^{2} + 136\,n^{2} x + 51\,n^{2} x^{2} - 45\,n^{3} - 76\,n^{3} x - 31\,n^{3} x^{2} + 11\,n^{4} + 20\,n^{4} x + 9\,n^{4} x^{2} - n^{5} - 2\,n^{5} x - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 10\,x^{2} + 4\,x^{3} - 17\,n x - 29\,n x^{2} - 12\,n x^{3} + 17\,n^{2} x + 30\,n^{2} x^{2} + 13\,n^{2} x^{3} - 7\,n^{3} x - 13\,n^{3} x^{2} - 6\,n^{3} x^{3} + n^{4} x + 2\,n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(27 + 24\,x - 63\,n - 56\,n x + 51\,n^{2} + 46\,n^{2} x - 17\,n^{3} - 16\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(6\,x + 6\,x^{2} - 11\,n x - 11\,n x^{2} + 6\,n^{2} x + 6\,n^{2} x^{2} - n^{3} x - n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3448
41524948
5721529616336
64201100975320252840
7288090001026050649603628200
822680823201146607565623275268849321328
920160083328013708801132992485520988327168644130336
1019958409253440175543201743638496049802844180397215184328159104872
11217728001118880002404080002789136001879920007351380015630300153368046080100931698360
122594592001463616000351267840046616169603718499400181235736052329354081972176574447511264012116039350272
133353011200205970688005460618240081581955840751932720004396056192016123282560353036217641494226421011880270336144299381921520
1446702656000310313203200900582883200149579958528015644957328001067753681280476263307520135071192256228617068682043395900754103356389761696021963393784
15697426329600498369231360015713886988800287223410073603361449611520026245508889600138096870912004830497992320108034486560014338118340098449751402664700081490944196129851375712968
Technical details
Node id
n-9b67fd9b7ecdcab9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-efe4fe8c256eb4ca

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       1;
   3:       4,       4;
   4:      15,      24,        9;
   5:      72,     152,       96,       16;
   6:     420,    1100,      975,      320,      25;
   7:    2880,    9000,    10260,     5064,     960,      36;
   8:   22680,   82320,   114660,    75656,   23275,    2688,     49;
   9:  201600,  833280,  1370880,  1132992,  485520,   98832,   7168,    64;
  10: 1995840, 9253440, 17554320, 17436384, 9604980, 2844180, 397215, 18432, 81;
%F Formula
The row polynomials satisfy (n^3-7*n^2+16*n-12)*P(n,x) = -(n^5-11*n^4+45*n^3-85*n^2+74*n-24+(2*n^5-20*n^4+76*n^3-136*n^2+114*n-36)*x+(n^5-9*n^4+31*n^3-51*n^2+40*n-12)*x^2)*P(n-2,x) + ((n^4-7*n^3+17*n^2-17*n+6)*x+(2*n^4-13*n^3+30*n^2-29*n+10)*x^2+(n^4-6*n^3+13*n^2-12*n+4)*x^3)*d/dx P(n-2,x) + (2*n^4-17*n^3+51*n^2-63*n+27+(2*n^4-16*n^3+46*n^2-56*n+24)*x)*P(n-1,x) - ((n^3-6*n^2+11*n-6)*x+(n^3-6*n^2+11*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 11*n^4 + 45*n^3 - 85*n^2 + 74*n - 24 + (2*n^5 - 20*n^4 + 76*n^3 - 136*n^2 + 114*n - 36)*x + (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*P[n - 2, x] + ((n^4 - 7*n^3 + 17*n^2 - 17*n + 6)*x + (2*n^4 - 13*n^3 + 30*n^2 - 29*n + 10)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 17*n^3 + 51*n^2 - 63*n + 27 + (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*P[n - 1, x] - ((n^3 - 6*n^2 + 11*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*D[P[n - 1, x], x])/(n^3 - 7*n^2 + 16*n - 12)]];
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; v[3] = 4 + 4*x; for(n = 4, N, my(i = n); v[i] = (-(n^5 - 11*n^4 + 45*n^3 - 85*n^2 + 74*n - 24 + (2*n^5 - 20*n^4 + 76*n^3 - 136*n^2 + 114*n - 36)*x + (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*v[i - 2] + ((n^4 - 7*n^3 + 17*n^2 - 17*n + 6)*x + (2*n^4 - 13*n^3 + 30*n^2 - 29*n + 10)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 17*n^3 + 51*n^2 - 63*n + 27 + (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*v[i - 1] - ((n^3 - 6*n^2 + 11*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*deriv(v[i - 1], x))/(n^3 - 7*n^2 + 16*n - 12)); 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. [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.