Real-rooted atlas

A008292 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(4\,x^{2} + 16\,n x - 28\,n x^{2} - n^{2} - 24\,n^{2} x + 53\,n^{2} x^{2} + 2\,n^{3} + 16\,n^{3} x - 38\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(x^{2} + 5\,x^{3} + 4\,n x + 4\,n x^{2} - 32\,n x^{3} - 5\,n^{2} x + 8\,n^{2} x^{2} + 53\,n^{2} x^{3} + 2\,n^{3} x - 12\,n^{3} x^{2} - 30\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{3} + x^{4} + 2\,n x^{2} - 4\,n x^{3} - 6\,n x^{4} - n^{2} x^{2} + 8\,n^{2} x^{3} + 9\,n^{2} x^{4} - 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-4\,x + n + 12\,n x - 3\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-x + 4\,n x - 4\,n x^{2} - 2\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
3624939
42414412616310
51209601350480252935
6720720014040864015503632186
750406048015120013440045150453649400855
84032056448017236802016000102060020865612446645586246
936288058060802095632030481920208530006713280889350326408186095551
10362880065318400272160000474163200410886000183708000401074803582720827821001453637482
1139916800798336000377213760076640256008062362000465230304014507724002238720001385148620460012126677781647
1247900160010538035200556839360001293304320001602992160001138347302404669844256010593792000118875794451915600495374144528698754662
1362270208001494484992008733396672002283240960000326140214400027573248102401411481111040429526056960729826297206078072000189939750117936016911251242090439
148717829120022666355712001451518548480042194292940800682896614400006708369507840041316052377600159116525568003699609273360480900420000301776274806816700802768558196255875725500474
15130767436800036614882304000254996501760000815988805632000147640068576000016542080755200001193619736108800559003808563200166864561783920302899156560003060335678400146382163200240835241464221362256192503780072295
Technical details
Node id
n-7016994b0a4cefab · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-856ad640314d7fe1

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): 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:       2,        4;
   3:       6,       24,         9;
   4:      24,      144,       126,        16;
   5:     120,      960,      1350,       480,        25;
   6:     720,     7200,     14040,      8640,      1550,        36;
   7:    5040,    60480,    151200,    134400,     45150,      4536,       49;
   8:   40320,   564480,   1723680,   2016000,   1020600,    208656,    12446,      64;
   9:  362880,  5806080,  20956320,  30481920,  20853000,   6713280,   889350,   32640,    81;
  10: 3628800, 65318400, 272160000, 474163200, 410886000, 183708000, 40107480, 3582720, 82782, 100;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^4-2*n^3+n^2+(4*n^4-16*n^3+24*n^2-16*n)*x-(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) + ((2*n^3-5*n^2+4*n)*x+(4*n^4-12*n^3+8*n^2+4*n+1)*x^2+(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) - ((n^2-2*n)*x^2+(4*n^3-8*n^2+4*n-1)*x^3+(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-3*n^2+n+(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) - ((2*n^2-4*n+1)*x+(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 2*n^3 + n^2 + (4*n^4 - 16*n^3 + 24*n^2 - 16*n)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] + ((2*n^3 - 5*n^2 + 4*n)*x + (4*n^4 - 12*n^3 + 8*n^2 + 4*n + 1)*x^2 + (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 - 2*n)*x^2 + (4*n^3 - 8*n^2 + 4*n - 1)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 3*n^2 + n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] - ((2*n^2 - 4*n + 1)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n^4 - 2*n^3 + n^2 + (4*n^4 - 16*n^3 + 24*n^2 - 16*n)*x - (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] + ((2*n^3 - 5*n^2 + 4*n)*x + (4*n^4 - 12*n^3 + 8*n^2 + 4*n + 1)*x^2 + (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 - 2*n)*x^2 + (4*n^3 - 8*n^2 + 4*n - 1)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 3*n^2 + n + (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] - ((2*n^2 - 4*n + 1)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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. [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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.