Real-rooted atlas

A039755 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-3 + 140\,x + 585\,x^{2} + 8\,n - 216\,n x - 1824\,n x^{2} - 4\,n^{2} + 144\,n^{2} x + 1976\,n^{2} x^{2} - 32\,n^{3} x - 896\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 4\,x^{2} + 612\,x^{3} + 120\,n x^{2} - 1752\,n x^{3} - 112\,n^{2} x^{2} + 1664\,n^{2} x^{3} + 32\,n^{3} x^{2} - 608\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(8\,x^{3} + 108\,x^{4} - 288\,n x^{4} + 240\,n^{2} x^{4} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(12 - 108\,x - 20\,n + 216\,n x + 8\,n^{2} - 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 36\,x^{2} - 120\,n x^{2} + 112\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
3116926
413611716170
5164522640251252
61100153052803025369972
7114435552432044275131044985448
811967119812003102753318845355764784296
9125612852220416145215034594562312212209920817667344
10132421492517440524055022279320352731401531008079712110079439568
11140033885109056015751050105289632308194810337807360976812212952400121868801440
12148450985211464041311050400149288188461457839606688003090443301605922900107173331449995993504
1315767385438368009748117512924679688965033924310375802884820493635127313457600367655087838263680169120629683264
1416761036626594016211386175367880713235352921604183260197120480306249423562923861500235021536750219152606401347202091961522811639104
151784141687108326404277523259462332880120476431075875888865280349550918016370856013228006365262388485197978062464012873468715946873086422520061623291008
Technical details
Node id
n-99d42afb2e5322c3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2bb878dfcdcd8bac

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: 1,   4;
   3: 1,  16,     9;
   4: 1,  36,   117,     16;
   5: 1,  64,   522,    640,      25;
   6: 1, 100,  1530,   5280,    3025,       36;
   7: 1, 144,  3555,  24320,   44275,    13104,       49;
   8: 1, 196,  7119,  81200,  310275,   331884,    53557,       64;
   9: 1, 256, 12852, 220416, 1452150,  3459456,  2312212,   209920,     81;
  10: 1, 324, 21492, 517440, 5240550, 22279320, 35273140, 15310080, 797121, 100;
%F Formula
The row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = -(4*n^2-8*n+3+(32*n^3-144*n^2+216*n-140)*x-(144*n^4-896*n^3+1976*n^2-1824*n+585)*x^2)*P(n-2,x) - (4*x-(32*n^3-112*n^2+120*n+4)*x^2-(64*n^4-608*n^3+1664*n^2-1752*n+612)*x^3)*d/dx P(n-2,x) + (8*x^3-(64*n^3-240*n^2+288*n-108)*x^4)*d^2/dx^2 P(n-2,x) + (8*n^2-20*n+12+(32*n^3-144*n^2+216*n-108)*x)*P(n-1,x) + (4*x-(32*n^3-112*n^2+120*n-36)*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[Together[(-(4*n^2 - 8*n + 3 + (32*n^3 - 144*n^2 + 216*n - 140)*x - (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*P[n - 2, x] - (4*x - (32*n^3 - 112*n^2 + 120*n + 4)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*D[P[n - 2, x], x] + (8*x^3 - (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*D[P[n - 2, x], {x, 2}] + (8*n^2 - 20*n + 12 + (32*n^3 - 144*n^2 + 216*n - 108)*x)*P[n - 1, x] + (4*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];
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 - 8*n + 3 + (32*n^3 - 144*n^2 + 216*n - 140)*x - (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*v[i - 2] - (4*x - (32*n^3 - 112*n^2 + 120*n + 4)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*deriv(v[i - 2], x) + (8*x^3 - (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*deriv(deriv(v[i - 2], x), x) + (8*n^2 - 20*n + 12 + (32*n^3 - 144*n^2 + 216*n - 108)*x)*v[i - 1] + (4*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 9)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.