Real-rooted atlas

A196347 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
2507818146
3750232212961624530
4150008330484888246241944209760
53750003557160591840029970005346002916013411320
611250000177493680454643280358279200108475200131220005248801123788240
739375000010175896080387348998404465887048020432412000411505920036006768011022480118881977760
81575000000066070734912036528926918405931431089920385062092352011487873600001650138739201093430016026453952015436402128000
97087500000004799617913664037947547491840084598842000960074929364072640031074848973504065175130621440701400083328036427091904071425670402406771499466880
10354375000000003860956830931200431821876151232001298226430488960001527595715237760008462850716880000024312734734982400377889413529600031605859152000013213749024000214277011200442710419175360000
11194906250000000034093363862194560053524562768506368002141769922082904960032850187095559680000236593374374206080008943862624520947200188308396894907520022552619715964800015081173176608000518550367104000707114136960094690642317096672000
12116943750000000000327997955577067776007187978395346740992003790861895065903257600747247533944402050560068608477204210944000003320242232114923315200909501252794865561600146274139493724211200139101135975648000007607133885415680002189225368028160025456108930560023266610131977788083200
137601343750000000000341573954425150487040010406728308020896051200071796998064384668275200017992606639246987587840002075246945547801343718400126032548849878475284480043690087049714656911360090615899448133784524800114912986968422010368008886190220914406400004048259547118026240098947895413086720099278824829184006500231872249863792844800
1453209406250000000000038288208539366854548480016169822655210743123404800145092942095562726160896000458378777201119865266176000656638092638221724110080000492985179009616346852198400211965587377910310776832000551398917368224724198400008953969593006925014528000918057866092178357145600588587397185084064768002268229929460688793600476736916829741568004169710642825728002046686909413330176978585600
Technical details
Node id
n-cc3dc35fe1aa407e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            5,              3;
  2:           50,             78,              18;
  3:          750,           2322,            1296,             162;
  4:        15000,          83304,           84888,           24624,            1944;
  5:       375000,        3557160,         5918400,         2997000,          534600,           29160;
  6:     11250000,      177493680,       454643280,       358279200,       108475200,        13122000,         524880;
  7:    393750000,    10175896080,     38734899840,     44658870480,     20432412000,      4115059200,      360067680,      11022480;
  8:  15750000000,   660707349120,   3652892691840,   5931431089920,   3850620923520,   1148787360000,   165013873920,   10934300160,    264539520;
  9: 708750000000, 47996179136640, 379475474918400, 845988420009600, 749293640726400, 310748489735040, 65175130621440, 7014000833280, 364270919040, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+3*n*x)*P(n-1,x) + 3*n*x*d/dx 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[(5*n + 3*n*x)*P[n - 1, x] + 3*n*x*D[P[n - 1, x], 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] = (5*n + 3*n*x)*v[i - 1] + 3*n*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.