Real-rooted atlas

A133214 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(26 + 16\,x + 19\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-15 - 22\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 61 + 64\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0616416141
1595900432641991
26145121608640256025629761
36552716042015136067840140801024460251
47129092081088245160014771204723207372840967272861
5786366726665828376881122857120012409600303206437273616384116619591
68761574533849088055820262451040460828080640094044160183500801835008655361889815041
798372669542654162608040695040861202944057416325122450165760660602880106168320884736026214430869546411

All 15 rows

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(-25 - 40\,x - 16\,x^{2} - 25\,n - 40\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 12\,x + 14\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 61 + 64\,x + 16\,x^{2}\)
\(P_{1}(x) = 595 + 900\,x + 432\,x^{2} + 64\,x^{3}\)
Technical details
Node id
n-0a43d0b65760bf28 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-0aff05df6ff0dde8)
Fingerprint
f6-20efcd8d91464382

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           61,           64,            16;
  1:          595,          900,           432,            64;
  2:         6145,        12160,          8640,          2560,           256;
  3:        65527,       160420,        151360,         67840,         14080,          1024;
  4:       712909,      2081088,       2451600,       1477120,        472320,         73728,         4096;
  5:      7863667,     26665828,      37688112,      28571200,      12409600,       3032064,       372736,        16384;
  6:     87615745,    338490880,     558202624,     510404608,     280806400,      94044160,     18350080,      1835008,       65536;
  7:    983726695,   4265416260,    8040695040,    8612029440,    5741632512,    2450165760,    660602880,    106168320,     8847360,     262144;
  8:  11112210781,  53436147520,  113351510160,  139195607040,  109084308480,   56648761344,  19565199360,   4376494080,   592773120,   41943040,   1048576;
  9: 126142119187, 666244968004, 1570688350000, 2175593555520, 1960099384320, 1200302788608, 506427457536, 145819238400, 27671592960, 3215196160, 196083712, 4194304;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (19*n+26+(12*n+16)*x)*P(n-1,x) - (15+22*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 61 + 64*x + 16*x^2.
%t Mathematica
P[0, x_] := 61 + 64*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 26 + (12*n + 16)*x)*P[n - 1, x] - (15 + 22*x + 8*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 61 + 64*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 26 + (12*n + 16)*x)*v[i - 1] - (15 + 22*x + 8*x^2)*deriv(v[i - 1], x))/(n + 2)); 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