Real-rooted atlas

A166979 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + x + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 11\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1111113
21301111143
3158681111111852
419523051405011111127563
511415820785902780611111111473725
611961230629393024785815394270111111119290396
71260230868600061335399174794538103602241111111111203745235
81333397262129736529142675688922052197849305197955369011111111114912490375
91415640354672200170049747300294848723246860890635193342303772263122111111111111128777672338
1015069806593467554690705471235568457216138249129292365318716018162639896617178411043101111111111113643086083981
1116061441111738808511513836624237619958084094514275283472477865923599143191458151573232814818136500920930011111111111111110557605978901
12171520471130502186257720663212642720690635530219158725459923063795241996483009781313832562136543151457700610907905259462860878130111111111111113579776914324250
1318332826465097328653529055583377581507301278220802001028199263325376034229225151851732070163839815371352638180516117560410750799662994704930905467795581111111111111111123074955978249433
1419603809408178170010450234158824872556508404783763798701215093818714490214704510434258132089623767333382281004913546831188358198868775906264919011550331445241807419369831499922715011111111111111114474133111905169219
15110965028601267326201936443945818698959723081155417790220984534320681863830110172299809583583158307124351259026012497844267164790898482147109132973136587473648022984220077132434626361676287898178037909609642696111111111111111111171363047274358839412
Technical details
Node id
n-78cee252b1c63b6a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8e4671726dd28243

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 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,  11,     1;
   2: 1,  30,   111,       1;
   3: 1,  58,   681,    1111,         1;
   4: 1,  95,  2305,   14050,     11111,           1;
   5: 1, 141,  5820,   78590,    278061,      111111,            1;
   6: 1, 196, 12306,  293930,   2478581,     5394270,      1111111,            1;
   7: 1, 260, 23086,  860006,  13353991,    74794538,    103602241,     11111111,             1;
   8: 1, 333, 39726, 2129736,  52914267,   568892205,   2197849305,   1979553690,     111111111,            1;
   9: 1, 415, 64035, 4672200, 170049747,  3002948487,  23246860890,  63519334230,   37722631221,   1111111111,           1;
  10: 1, 506, 98065, 9346755, 469070547, 12355684572, 161382491292, 923653187160, 1816263989661, 717841104310, 11111111111, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(9*n+1)*x)*P(n-1,x) - 9*x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 11*x + x^2.
%t Mathematica
P[1, x_] := 1 + 11*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (9*n + 1)*x)*P[n - 1, x] - 9*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 11*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + (9*n + 1)*x)*v[i - 1] - 9*x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.