Real-rooted atlas

A038320 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(34 + 48\,x + 17\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 816 + 1152\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
081611521968
1208085875241472121032
24716481997568282009613271046616416
3100225205659776011985408011280384039813120339091320
42044594081443242880407503872057529958404060938240114661785616683292944
540551115923434918054412123240192022820216832024162582528013644752486432105299968798017512488
67878502521677858142566432975213322247758873722880109537040793609278431690752436632079564888060251340837392820585152
715067636072561701756544665684086793971712237421535920128418979181035520473200016228352334023540867072134732184551424237762678620161724743849490136

All 15 rows

Technical details
Node id
n-1bca2d435470a24f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3d4815a526d230d8

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             816,             1152;
  1:           20808,            58752,             41472;
  2:          471648,          1997568,           2820096,            1327104;
  3:        10022520,         56597760,         119854080,          112803840,           39813120;
  4:       204459408,       1443242880,        4075038720,         5752995840,         4060938240,         1146617856;
  5:      4055111592,      34349180544,      121232401920,       228202168320,       241625825280,       136447524864,        32105299968;
  6:     78785025216,     778581425664,     3297521332224,      7758873722880,     10953704079360,      9278431690752,      4366320795648,       880602513408;
  7:   1506763607256,   17017565446656,    84086793971712,    237421535920128,    418979181035520,    473200016228352,    334023540867072,    134732184551424,     23776267862016;
  8:  28461090359280,  361623265741440,  2042107853598720,   6726943517736960,  14245292155207680,  20111000689704960,  18928000649134080,  11452235686871040,   4041965536542720,    634033809653760;
  9: 532222389718536, 7513727854849920, 47734271077870080, 179705491116687360, 443978272170639360, 752151425794965504, 884884030347018240, 713856024481628160, 377923777666744320, 118564322405253120, 16738492574859264;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(24*n+48)*x)*P(n-1,x) for n >= 1, with P(0,x) = 816 + 1152*x.
%t Mathematica
P[0, x_] := 816 + 1152*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (24*n + 48)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 816 + 1152*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (24*n + 48)*x)*v[i - 1])/(n + 1)); 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.