Real-rooted atlas

A178821 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(24 + 16\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(16\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1302050
25409602401740
375602744016800224054040
490720609280689920215040179201622880
5979776116202242177280011934720225792012902448694464
69797760200202240585500160503193600162355200206438408601601482552960
7923788803207621120141394176001791347712087757824001863966720170311680540672046168362240
883140992048668590080316173496320568500387840397081681920125349396480188478259201297612800324403201476782841600
972055526407077944473606674927001600166077902848001590304235520070018159411201540980080640172625756160927793152018743296048625646784000
106052664217699494733578241347922181365764558959063859205828173301350403439231374458881039313326571521684647444480014602901913606297747456010496245761649740716495872
1149521798144013600478201856026275758372864001192540252839936019971224847974400153726421283635206127680722042880134796199329792016755944325120011572110950400409353584640572522496057688534690375680
12396174385152018160382561484804977344113344512030022339848044544064942474235871232063943145963913216032709465356632064093788574213734400156788090889830401541688577228800868249449267202564900782080305345331202078866187438243840
133108445175808023768915548241920920736122398310400732715030246850560020253849595294515200251376605877606809601617689327494496256058809409490806374401269702593693614080166686512476323840132870715526348806218663015219201557261189120015971909632077171344608302694400
1423979434213376030576020565786624016695490340274831360174306407058033868800610577090108168601600944478737566657413120753245280770646343680340276584045241958400922788801728741376001553097072700293120016430802959912140801083106302256742404279353747701760919981994803208214124953602949451203888476651520
Technical details
Node id
n-f1b46426574116be · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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:         30,           20;
  2:        540,          960,           240;
  3:       7560,        27440,         16800,           2240;
  4:      90720,       609280,        689920,         215040,          17920;
  5:     979776,     11620224,      21772800,       11934720,        2257920,        129024;
  6:    9797760,    200202240,     585500160,      503193600,      162355200,      20643840,        860160;
  7:   92378880,   3207621120,   14139417600,    17913477120,     8775782400,    1863966720,     170311680,      5406720;
  8:  831409920,  48668590080,  316173496320,   568500387840,   397081681920,  125349396480,   18847825920,   1297612800,   32440320;
  9: 7205552640, 707794447360, 6674927001600, 16607790284800, 15903042355200, 7001815941120, 1540980080640, 172625756160, 9277931520, 187432960;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n+24+(4*n+16)*x)*P(n-1,x) + (4*n+16)*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[Together[((6*n + 24 + (4*n + 16)*x)*P[n - 1, x] + (4*n + 16)*x*D[P[n - 1, x], x])/(n)]];
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] = ((6*n + 24 + (4*n + 16)*x)*v[i - 1] + (4*n + 16)*x*deriv(v[i - 1], x))/(n)); 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.