Real-rooted atlas

A038321 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(36 + 28\,x + 18\,n + 14\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 504 + 392\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0504392896
11360821168823243008
23265927620485927041536641835008
37348320228614402667168013829760268912073400320
4158723712617258880960180480746807040290424960451772162818572288
53333197952155549237763024568512031365895680182967724805692329216737894528105226698752
6685686435843733181706248710757314561129172244480878245079040409847703552106256812032118063124483848290697216
7138851503257686396490915842351904474931236585180721152355689257011202213177599180886068017745921912622616576185949421056138538465099776

All 15 rows

Technical details
Node id
n-f29c7367d5544e24 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-27210632b0078bca

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:             504,              392;
  1:           13608,            21168,              8232;
  2:          326592,           762048,            592704,            153664;
  3:         7348320,         22861440,          26671680,          13829760,           2689120;
  4:       158723712,        617258880,         960180480,         746807040,         290424960,          45177216;
  5:      3333197952,      15554923776,       30245685120,       31365895680,       18296772480,        5692329216,         737894528;
  6:     68568643584,     373318170624,      871075731456,     1129172244480,      878245079040,      409847703552,      106256812032,       11806312448;
  7:   1388515032576,    8639649091584,    23519044749312,    36585180721152,    35568925701120,    22131775991808,     8606801774592,     1912622616576,     185949421056;
  8:  27770300651520,  194392104560640,   604775436410880,  1097555421634560,  1280481325240320,   995929919631360,   516408106475520,   172136035491840,   33470895790080,   2892546549760;
  9: 549851952900096, 4276626300334080, 14968192051169280, 31045139069091840, 42255883732930560, 39438824817401856, 25562201270538240, 11360978342461440, 3313618683217920, 572724216852480, 44545216866304;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (18*n+36+(14*n+28)*x)*P(n-1,x) for n >= 1, with P(0,x) = 504 + 392*x.
%t Mathematica
P[0, x_] := 504 + 392*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 36 + (14*n + 28)*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] = 504 + 392*x; for(n = 1, N, my(i = n + 1); v[i] = ((18*n + 36 + (14*n + 28)*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.