Real-rooted atlas

A108838 → D → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(2 - 4\,x + 6\,n + n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
212416
356426104
4240288968632
59901650900180103730
640048580660022003001221696
716016420424204220020462046214125216
86364819801624460815288050960873667216719536
92519409069841336608103958445864011466015288936184124658
10994840406980069768006511680359856011995202352002520012602023612880
1139225121797818035174700383724002558160010744272282744044880039600165022135091176
1215452320784502401725905282157381601688385608595417628651392613958480784059840211224772684776
13608435103389852708286306601168581700105172353063103411825607181669837768124710301385670875162652264419655266
1423951970014530861803908300760614161548062553491004330626300207870062469290020815816200423963940228228012448832762825284634368
Technical details
Node id
n-bc5af0fb8df84692 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-033fc5829f38f9ac

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:      2;
  2:     12,      4;
  3:     56,     42,       6;
  4:    240,    288,      96,       8;
  5:    990,   1650,     900,     180,     10;
  6:   4004,   8580,    6600,    2200,    300,     12;
  7:  16016,  42042,   42042,   20020,   4620,    462,    14;
  8:  63648, 198016,  244608,  152880,  50960,   8736,   672,  16;
  9: 251940, 906984, 1336608, 1039584, 458640, 114660, 15288, 936, 18;
%F Formula
The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (4*n^2+6*n+2+(3*n^2+n-4)*x)*P(n-1,x) - ((2*n+2)*x+(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 6*n + 2 + (3*n^2 + n - 4)*x)*P[n - 1, x] - ((2*n + 2)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 6*n + 2 + (3*n^2 + n - 4)*x)*v[i - 1] - ((2*n + 2)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); 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

Same fingerprint ?

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