Real-rooted atlas

A008957 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} x + 12\,n^{3} x^{2} - 2\,n^{4} x - 13\,n^{4} x^{2} + 6\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-6\,n^{2} x^{2} - 5\,n^{2} x^{3} + 2\,n^{3} x^{2} + 12\,n^{3} x^{3} - 9\,n^{4} x^{3} + 2\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(-n^{2} x^{3} - n^{2} x^{4} + 2\,n^{3} x^{4} - n^{4} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 2\,x - 2\,n - 8\,n x + 2\,n^{2} + 12\,n^{2} x - 8\,n^{3} x + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(4\,x^{2} - 10\,n x^{2} + 8\,n^{2} x^{2} - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x) + \left(x^{3} - 2\,n x^{3} + n^{2} x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2527
31442359
4302942554583
55512544224136456902
691400434320520526825697298
7140105561844707842125924103276671604561
8204242767544946933212162766506430794152915830572553
928550388253878043148924232939525318942474676762526990489665995685
1038596558737602820901588421746411057307128224602126910069679764031457251016419470659
11506173558191078258384369401497686400010125162396021914584230011080000404070582218301398101011454104255980
126502960104514152529020820408205181920098189122716044717140486206372109784040200145829455070596199310615164551213982830409572
138194832109886734089209520403758876257507283470941720606031625346001903512020701201812170459706753565207699131069896388962426843545213476191098762673
14101576003220324070024871549560149100663075043819748164620604435659516360358412830400772078904407298105355069458909510060628175518936680686282298812411632203051417828838938336475
Technical details
Node id
n-ee02330825df0350 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a07a15ae93e03f31

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   1;
  2:   5,     2;
  3:  14,    42,       3;
  4:  30,   294,     255,        4;
  5:  55,  1254,    4224,     1364,         5;
  6:  91,  4004,   34320,    52052,      6825,         6;
  7: 140, 10556,  184470,   784212,    592410,     32766,        7;
  8: 204, 24276,  754494,  6933212,  16276650,   6430794,   152915,      8;
  9: 285, 50388, 2538780, 43148924, 232939525, 318942474, 67676252, 699048, 9;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-8*n^3+8*n^2)*x+(n^6-6*n^5+13*n^4-12*n^3+4*n^2)*x^2)*P(n-2,x) + ((2*n^3-6*n^2)*x^2+(2*n^5-9*n^4+12*n^3-5*n^2)*x^3)*d/dx P(n-2,x) - (n^2*x^3+(n^4-2*n^3+n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-2*n+1+(2*n^4-8*n^3+12*n^2-8*n+2)*x)*P(n-1,x) - (2*n^3-8*n^2+10*n-4)*x^2*d/dx P(n-1,x) + (n^2-2*n+1)*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*P[n - 2, x] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*D[P[n - 2, x], x] - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*P[n - 1, x] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*D[P[n - 1, x], x] + (n^2 - 2*n + 1)*x^3*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*v[i - 2] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*deriv(v[i - 2], x) - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*v[i - 1] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*deriv(v[i - 1], x) + (n^2 - 2*n + 1)*x^3*deriv(deriv(v[i - 1], x), 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 - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.