Real-rooted atlas

A038320 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(253 + 204\,x + 36\,x^{2} + 66\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 253 + 204\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
025320436493
135094554183621610115
2465858421654648146881296201433
36002811397550126324054648011016077763925487
4756939721610116251559001515888049183207931524665675252421
5938927333179146744538124363521826001591682404131388855520642799361424116571
611497430894506851184762995217672609989764226191200152801510433051110438071296167961626672013745
71393332726562086126806121684981968137339139168980234861764564286496013752135936254965708825698124810077696495278778311

All 15 rows

Technical details
Node id
n-387f8733bacab0a3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-efae2a5576a476b8

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           253,            204,             36;
  1:          3509,           4554,           1836,            216;
  2:         46585,          84216,          54648,          14688,           1296;
  3:        600281,        1397550,        1263240,         546480,         110160,           7776;
  4:       7569397,       21610116,       25155900,       15158880,        4918320,         793152,          46656;
  5:      93892733,      317914674,      453812436,      352182600,      159168240,       41313888,        5552064,        279936;
  6:    1149743089,     4506851184,     7629952176,     7260998976,     4226191200,     1528015104,      330511104,      38071296,      1679616;
  7:   13933327265,    62086126806,   121684981968,   137339139168,    98023486176,    45642864960,    13752135936,    2549657088,    256981248,     10077696;
  8:  167414286061,   835999635900,  1862583804180,  2433699639360,  2060087087520,  1176281834112,   456428649600,  117875450880,  19122428160,   1713208320,    60466176;
  9: 1997181694277, 11049342880026, 27587987984700, 40976843691960, 40156044049440, 27193149555264, 12939100175232, 4303470124800, 972472469760, 140231139840, 11307174912, 362797056;
%F Formula
The row polynomials satisfy (6*n+17+6*x)*P(n,x) = (66*n+253+(36*n+204)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 253 + 204*x + 36*x^2.
%t Mathematica
P[0, x_] := 253 + 204*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((66*n + 253 + (36*n + 204)*x + 36*x^2)*P[n - 1, x])/(6*n + 17 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 253 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((66*n + 253 + (36*n + 204)*x + 36*x^2)*v[i - 1])/(6*n + 17 + 6*x)); 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. [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. [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.