Real-rooted atlas

A156992 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
215023454438
330009288518464818120
47500041652042444012312097201048800
52250000213429603551040017982000320760017496080467920
6787500001242455760318250296025079544007593264009185400036741607866517680
731500000008140716864030987919872035727096384016345929600032920473600288054144088179840951055822080
81417500000005946366142080328760342265605338287980928034655588311680103390862400001485124865280984087014402380855680138927619152000
9708750000000047996179136640037947547491840008459884200096000749293640726400031074848973504006517513062144007014000833280036427091904007142567040024067714994668800
10389812500000000424705251402432004750040637663552001428049073537856000168035528676153600093091357885680000026744008208480640041567835488256000347664450672000014535123926400023570471232004869814610928960000
11233887500000000004091203663463347200642294753222076416002570123906499485952003942022451467161600002839120492490472960001073263514942513664002259700762738890240027063143659157760001809740781192960006222604405248000848536964352001136287707805160064000
12152026875000000000042639734225018810880093443719139507632896004928120463585674234880097142179412772266572800891910203654742272000004316314901749400309760011823516286333252300800190156381341841474560018083147676834240000098892740510403840002845992978436608003309294160972800302465931715711245081600
131064188125000000000004782035361952106818560014569419631229254471680001005157972901385355852800025189649294945782622976000290534572376692188120576001764455683898298653982720061166121869600519675904001268622592273872983347200160878181755790814515200124406663092801689600005667563365965236736001385270535783214080013899035476085760091003246211498093099827200
1479814109375000000000005743231280905028182272000242547339828161146851072000217639413143344089241344000068756816580167979789926400009849571389573325861651200000739477768514424520278297600031794838106686546616524800008270983760523370862976000001343095438951038752179200001377086799138267535718400088288109577762609715200034023448941910331904000715105375244612352000625456596423859200030700303641199952654678784000
Technical details
Node id
n-8f062cfeefb3b819 · 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:            10,               6;
  2:           150,             234,               54;
  3:          3000,            9288,             5184,              648;
  4:         75000,          416520,           424440,           123120,             9720;
  5:       2250000,        21342960,         35510400,         17982000,          3207600,           174960;
  6:      78750000,      1242455760,       3182502960,       2507954400,        759326400,         91854000,         3674160;
  7:    3150000000,     81407168640,     309879198720,     357270963840,     163459296000,      32920473600,      2880541440,       88179840;
  8:  141750000000,   5946366142080,   32876034226560,   53382879809280,   34655588311680,   10339086240000,   1485124865280,    98408701440,    2380855680;
  9: 7087500000000, 479961791366400, 3794754749184000, 8459884200096000, 7492936407264000, 3107484897350400, 651751306214400, 70140008332800, 3642709190400, 71425670400;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+5+(3*n+3)*x)*P(n-1,x) + (3*n+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[(5*n + 5 + (3*n + 3)*x)*P[n - 1, x] + (3*n + 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] = (5*n + 5 + (3*n + 3)*x)*v[i - 1] + (3*n + 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

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.