Real-rooted atlas

A038337 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(529 + 1012\,x + 363\,x^{2} + 253\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 1012\,x + 363\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
052910123631904
1121673491425047532477452
227984110706961152162489808732053065712
364363433078251044166210281639608418575966306118933904
414803588984959727615237342451295542160580881675133350228124009274543542400
5340482544722797526906490642426895214557194031173983225107346933541996549247155897368171473290176
6783109852815992492786721504636775796191895704739214340032283506583945257121836825307242868511571219292299296407848717568
71801152661463155055750856384449425894139666204018135024593677336536903407191670559612674094619956296890947619239935059530323579476910237510589351168

All 15 rows

Technical details
Node id
n-1ffda96ef41ce683 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4aefc355b365f862

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:             529,              1012,               363;
  1:           12167,             34914,             25047,              5324;
  2:          279841,           1070696,           1152162,            489808,             73205;
  3:         6436343,          30782510,          44166210,          28163960,           8418575,            966306;
  4:       148035889,         849597276,        1523734245,        1295542160,         580881675,         133350228,          12400927;
  5:      3404825447,       22797526906,       49064242689,       52145571940,       31173983225,       10734693354,        1996549247,         155897368;
  6:     78310985281,      599249278672,     1504636775796,     1918957047392,     1434003228350,      658394525712,      183682530724,       28685115712,       1929229929;
  7:   1801152661463,    15505575085638,    44494258941396,    66204018135024,    59367733653690,    34071916705596,    12674094619956,     2968909476192,     399350595303,     23579476910;
  8:  41426511213649,   396253585521860,  1279209944565135,  2175274881579360,  2275763123391450,  1567308168457416,   728760440647470,   227616393174720,   45925318459845,   5423279689300,   285311670611;
  9: 952809757913927, 10025215713703058, 35960012886108795, 68793068129947260, 82252581459719550, 66088161103287708, 36875278296761982, 14396736868301040, 3873035190113595, 686044880696450, 72183852664583, 3423740047332;
%F Formula
The row polynomials satisfy (23+(11*n+22)*x)*P(n,x) = (529+(253*n+1012)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 1012*x + 363*x^2.
%t Mathematica
P[0, x_] := 529 + 1012*x + 363*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((529 + (253*n + 1012)*x + (121*n + 363)*x^2)*P[n - 1, x])/(23 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 529 + 1012*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((529 + (253*n + 1012)*x + (121*n + 363)*x^2)*v[i - 1])/(23 + (11*n + 22)*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

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.