Real-rooted atlas

A008459 → x ↦ x+1 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} - n - 6\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 3\,x + 6\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1336
21324946
36317113527396
432111521404648813606
516837515124209180283524333876
68989480241003051026005224511664729324556
74863930254776412710007557172552704594592721873151896
8265729188620855893608932896828954044634241306368174960656130915046
91462563116659713967963274907504855058686019498825474176598363264953919683305543556
10809745371700120275301585599646240813384180709357824395890740135943920262768052361960590493038019876
1145046719438439419187570201546310838757281840060758716081252789718522409067980687428775111504195844400717714730354866856
12251595969266978419212592798812347586584406218441806575425815872630054754563605201265613766979105332563968045987361229760696531441304523343996
13140993361916199913483835194321002548757142365113334461957080978517476912526832644769925732002295841063237468724191515503196852185154044410363099515943233065412858696
147923848253980047096565482771937371833110088264407642941005363483037057447097823685953574428157617633478601926771379345431464387807909489700269176887302530781357128352478296930946859111256
Technical details
Node id
n-862e5f3c5e2d148f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-3d3f3d7068653107

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 0)
  • 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:       3,        3;
  2:      13,       24,        9;
  3:      63,      171,      135,       27;
  4:     321,     1152,     1404,      648,       81;
  5:    1683,     7515,    12420,     9180,     2835,      243;
  6:    8989,    48024,   100305,   102600,    52245,    11664,      729;
  7:   48639,   302547,   764127,  1000755,   717255,   270459,    45927,    2187;
  8:  265729,  1886208,  5589360,  8932896,  8289540,  4463424,  1306368,  174960,   6561;
  9: 1462563, 11665971, 39679632, 74907504, 85505868, 60194988, 25474176, 5983632, 649539, 19683;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-1+(6*n-6)*x+(9*n-9)*x^2)*P(n-2,x) + (6*n-3+(6*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*P[n - 2, x] + (6*n - 3 + (6*n - 3)*x)*P[n - 1, x])/(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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n - 1 + (6*n - 6)*x + (9*n - 9)*x^2)*v[i - 2] + (6*n - 3 + (6*n - 3)*x)*v[i - 1])/(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

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