Real-rooted atlas

A227450 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-8\,x + 16\,x^{2} - 8\,x^{3} + 2\,x^{4} - 4\,n + 8\,n x - 12\,n x^{2} + 7\,n x^{3} - 2\,n x^{4} - 4\,n^{2} x + 3\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3} - 2\,x^{4} + 2\,x^{5} + 2\,n x^{2} - n x^{3} + 3\,n x^{4} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 10\,x - 6\,x^{2} + 2\,x^{3} + 4\,n - 4\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x^{2} - 2\,x^{3} - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 6 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1628
22016238
364104342204
41925603806821202
554425603200121613427656
614721030421280152323668264252222
738403763211782414246465856107765222378916
89728127488567168105907284403226952031340103622909386
92406440704024499206596352826492846348801068160909922062223538400
10583681239040971520035802624656261765885932824219360415606426483641122199945110
1113926436270083596032017424844844224435258858905639369932812241292816013888774088821021777716892
1232768010276864125785088776528896261556838448757463044907522048251874022460495635261450480227318016404216499191906
13761856283279364197007363218575360139167508483475051008049552588800387935708161560153795229450803202356972806704928327902159469839704
14175308876267520134557696012555984896678627389442191852748804225668761604740078520322946382158729436978387214199188640905704896198527246556021601735136046
Technical details
Node id
n-69c62b7596a70c08 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6573753d5e56d980

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:     2;
  1:     6,      2;
  2:    20,     16,       2;
  3:    64,    104,      34,       2;
  4:   192,    560,     380,      68,       2;
  5:   544,   2560,    3200,    1216,     134,       2;
  6:  1472,  10304,   21280,   15232,    3668,     264,       2;
  7:  3840,  37632,  117824,  142464,   65856,   10776,     522,     2;
  8:  9728, 127488,  567168, 1059072,  844032,  269520,   31340,  1036,    2;
  9: 24064, 407040, 2449920, 6596352, 8264928, 4634880, 1068160, 90992, 2062, 2;
%F Formula
The row polynomials satisfy (n-1+2*x)*P(n,x) = -(4*n+(4*n^2-8*n+8)*x+(n^3-3*n^2+12*n-16)*x^2+(3*n^2-7*n+8)*x^3+(2*n-2)*x^4)*P(n-2,x) + (2*n*x^2+(n^2-n+4)*x^3+(3*n-2)*x^4+2*x^5)*d/dx P(n-2,x) + (4*n-2+(2*n^2-4*n+10)*x+(5*n-6)*x^2+2*x^3)*P(n-1,x) - ((n-1)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 6 + 2*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + (4*n^2 - 8*n + 8)*x + (n^3 - 3*n^2 + 12*n - 16)*x^2 + (3*n^2 - 7*n + 8)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (2*n*x^2 + (n^2 - n + 4)*x^3 + (3*n - 2)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (4*n - 2 + (2*n^2 - 4*n + 10)*x + (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] - ((n - 1)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + (4*n^2 - 8*n + 8)*x + (n^3 - 3*n^2 + 12*n - 16)*x^2 + (3*n^2 - 7*n + 8)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (2*n*x^2 + (n^2 - n + 4)*x^3 + (3*n - 2)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (4*n - 2 + (2*n^2 - 4*n + 10)*x + (5*n - 6)*x^2 + 2*x^3)*v[i - 1] - ((n - 1)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.