Real-rooted atlas

A143491 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 5\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
015612
119262460
211471154120360
312015558010447202520
4127295166551048028504020160
5135511402518424488606926440320181440
61448268624546492146765090046636963628801814400
7154126616884140889761166265576457537366999840362880019958400
816518603081032661323109451102859034967140702909368062704039916800239500800
917726405313069633362303013875993016731022048989661692411827210074412804790016003113510400
10190364187450138702315282630119753843671189310266492947672927742801302097881613575738240622702080043589145600
11110449011384242611323347009523337106632347743112120615798164460178694411511981873619586944166419628735616087178291200653837184000
1211196461211939468768373870797854224943735340305747277726496225525484184784146622896192266672270431343289811203031488633600130767436800010461394944000
1311358365315315807870714887372520361576952102100214516493217540898196910812043925543698401446901268904033896996544384531807523315204981149250560020922789888000177843714048000
1411521066045752013439062286211744456701102055635682960522289211873378581609005621086029207880891424369763202798702122580646294296935860489538842821414408677181624832003556874280960003201186852864000
Technical details
Node id
n-c4e95af1e3477dcf · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2ee8a94f56aa5e4f

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: 1,  5,    6;
  1: 1,  9,   26,    24;
  2: 1, 14,   71,   154,    120;
  3: 1, 20,  155,   580,   1044,     720;
  4: 1, 27,  295,  1665,   5104,    8028,     5040;
  5: 1, 35,  511,  4025,  18424,   48860,    69264,     40320;
  6: 1, 44,  826,  8624,  54649,  214676,   509004,    663696,    362880;
  7: 1, 54, 1266, 16884, 140889,  761166,  2655764,   5753736,   6999840,   3628800;
  8: 1, 65, 1860, 30810, 326613, 2310945, 11028590,  34967140,  70290936,  80627040,   39916800;
  9: 1, 77, 2640, 53130, 696333, 6230301, 38759930, 167310220, 489896616, 924118272, 1007441280, 479001600;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 5*x + 6*x^2.
%t Mathematica
P[0, x_] := 1 + 5*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 3)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 5*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n + 3)*x)*v[i - 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.