Real-rooted atlas

A367270 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(8 + 6\,x - 21\,n - 15\,n x + 13\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x) + \left(4 + 7\,x + 3\,x^{2} - 8\,n - 14\,n x - 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
29615
3739027190
459310445941082339
54881111009180324040528806
64066511313012231063720157951458357078
73421051124550149971510319403827257144251034457580
82900385109947601742958014900760736533020820243061801749655996515
9247429451062342001950933601993118401231864204678430410532592125971259049707204422
1021215504910174164302123692020252355824018726989408944130162725002005038848050191651968308972038370
1118267848739677053650226252921953065578164026557270740152946707765890597020148383576023058634519486170649539114262008708
12157857336179153664066823695131339036053849898035717307615024093451166411233291053635941158000765481711510176111007404744621257641459942444430
1313682290564986196289842024471601628764131313397640460696213048535632733788081951879129200758744692992206160430905377728579804357107612276349320690873318706692350620
14118901878872980862979453622498212820212246337903716104574442697178355012207998597831558733089884144541584570244787574644853112364401149017957025286218183785256101558375122320522240284600501772

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-18 - 36\,x - 18\,x^{2} + 45\,n + 90\,n x + 45\,n x^{2} - 18\,n^{2} - 36\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(18 + 12\,x - 40\,n - 24\,n x + 20\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-4b3c71e576f04d69 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-b3a592a78406eb38)
Fingerprint
f6-4cbc0e728e3582be

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        1;
  2:        9,         6;
  3:       73,        90,        27;
  4:      593,      1044,       594,       108;
  5:     4881,     11100,      9180,      3240,       405;
  6:    40665,    113130,    122310,     63720,     15795,     1458;
  7:   342105,   1124550,   1499715,   1031940,    382725,    71442,     5103;
  8:  2900385,  10994760,  17429580,  14900760,   7365330,  2082024,   306180,   17496;
  9: 24742945, 106234200, 195093360, 199311840, 123186420, 46784304, 10532592, 1259712, 59049;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (13*n^2-21*n+8+(9*n^2-15*n+6)*x)*P(n-1,x) - (8*n-4+(14*n-7)*x+(6*n-3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n^2 - 21*n + 8 + (9*n^2 - 15*n + 6)*x)*P[n - 1, x] - (8*n - 4 + (14*n - 7)*x + (6*n - 3)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((13*n^2 - 21*n + 8 + (9*n^2 - 15*n + 6)*x)*v[i - 1] - (8*n - 4 + (14*n - 7)*x + (6*n - 3)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023