Real-rooted atlas

A088996 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
210271855
380306387162935
4880432679296426194421505
5123207376417589620889912360629160623645
62094401475748431798467174115861484272062852488021827575
74188800339132001173503882250258842582953111775037246763078811022480894930575
8963424008805348003512975724799200464411341413369102816731165815597500187665595226453952042061737025
92504902400254951496001151118084243026424652925106608729825735416619794288107091322058141872525754773822471425670402229272062325
10726421696008145064104004103096932296122299857461882388843927523831952534386851296417604241981883293270426878433100260561933566590880214277011200131527051677175
1123245494272002846139672960015817781337667252676174264378411680195864318201810799596462086200296996834041915808319405351108724727000746363207033617680087066456185744070711413696008549258359016375
12813592299520001079832664915200656083375044912024131062274092632598441082602899241054266909877185301352927343645497611274360367789839348744649436187622042633634864567176140185806886586882791414174803840254561089305600606997343490162625
13309165073817600044206651234905600291425156448759360117285288268303569632151875425806297606340134479684611176925276485437391358810118986037818830171829297222013166978650304914049667252682195417825887150008652798385500234752118538474210962560992788248291840046738795448742522125
14126757680265216000194232203163452160013805110766265168960603268247608523566561810825103184933193923949833904554555081366456490071760841265008034945514342764077077650092746137893284085545549808491658670003012927698699756115841189722824927699251683227760963365932454453856590986600819204169710642825728003879320022245629336375
Technical details
Node id
n-70ec15c0fd6aee57 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c19c84430e4d4959

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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;
  1:          2,           3;
  2:         10,          27,           18;
  3:         80,         306,          387,          162;
  4:        880,        4326,         7929,         6426,         1944;
  5:      12320,       73764,       175896,       208899,       123606,        29160;
  6:     209440,     1475748,      4317984,      6717411,      5861484,      2720628,       524880;
  7:    4188800,    33913200,    117350388,    225025884,    258295311,    177503724,     67630788,     11022480;
  8:   96342400,   880534800,   3512975724,   7992004644,  11341413369,  10281673116,   5815597500,   1876655952,   264539520;
  9: 2504902400, 25495149600, 115111808424, 302642465292, 510660872982, 573541661979, 428810709132, 205814187252, 57547738224, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-1+3*n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n - 1 + 3*n*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n - 1 + 3*n*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

Candidate equalities ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.