Real-rooted atlas

A130534 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x + n\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
23042981
321051024327990
416806240493211888114121
5151207992092700361805265243229428
615120010843201722960965520224370218707294170969
716632001563408032543280244792808057070124173086751218783707578
81995840023950080063268128061169472026680752057697920632772033242465611835007345
925945920038918880001273944672015355327680850527216024159643203692530803028557612400291968343568156448
10363242880066940473600266464598400391390876800267203059200955787817601906399152021677544001379778304527090590491112584528449
1154486432000121582581120057965779872001019277483840083886552288003668972630400915848065440135231132960118843985706040712701623847517714730380877011862
128717829120002326414970880013117219315200027220910492160026546067092160013890411722880042145752116160775109611584088106934168061628260320255918366057395628531441882724181789529
13148203095040004678983429120003086752891622400747095330741760085149867714912005241313764086400189181367575200042097890875040059377131356400535485147912030516641298010547332380200353257159432327174565870064460
1426676557107200098703261296640007548576442767360021097599797514240027786246306698880019850831452932480083829233619331200220936184685216003756776243220000419099710629600307125422604001453413190320424610730186919361824782969883100568854810889
Technical details
Node id
n-214429b77af4dcc0 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-af5fbb12372ecdb6

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:        30,         42,           9;
  3:       210,        510,         243,          27;
  4:      1680,       6240,        4932,        1188,         81;
  5:     15120,      79920,       92700,       36180,       5265,        243;
  6:    151200,    1084320,     1722960,      965520,     224370,      21870,       729;
  7:   1663200,   15634080,    32543280,    24479280,    8057070,    1241730,     86751,     2187;
  8:  19958400,  239500800,   632681280,   611694720,  266807520,   57697920,   6327720,   332424,    6561;
  9: 259459200, 3891888000, 12739446720, 15355327680, 8505272160, 2415964320, 369253080, 30285576, 1240029, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (n+4+3*x)*P(n-1,x) + 3*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[(n + 4 + 3*x)*P[n - 1, x] + 3*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] = (n + 4 + 3*x)*v[i - 1] + 3*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

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.