Real-rooted atlas

A085880 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-30 - 54\,x + 20\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
25914
350180162392
462533756075364513720
587506300017010020412091854537824
61312501181250425250076545006889050248005822588608
720625002227500010023750024057000032476950023383404070150212993898752
8335156254222968752280403125684120937512314176875132993110257979586615205189370145222393216
9558593750804375000050675625000182432250000410472562500591080490000531972441000273585826800615568110302110378350080
109496093750153836718750110762437500046520223750001256046041250022608828742500271305944910002092931575020094181920875901883638417518100454009463808
1116402343750029524218750002391461718750011479016250000036158901187500078103226565000011715483984750001205021209860000813389316655500325355726662200585640307991964858321184976896
12287041015625056834121093750511507089843750276213828515625099436978265625002505811852293750045104613341287500579916457245125005219248115206125031315488691236750112735759288452301844766970174674238057738063867904
135078417968750010969382812500001085968898437500065158133906250000263890442320312500760004473882500000159600939515325000024624144953793000002770216307301712500221617304584137000011967334447543398003916582182832384805874873274248577211793014101010071552
1490686035156250021220532226562500229181748046875000151259953710937500068066979169921875002205370125105468750052928883002531250000952719894045562500001286171856961509375001286171856961509375009260437370122867500045460328907875895000136380986723627685001888352123865614100589650705050503577600
151632348632812500041135185546875000048128167089843750003465228030468750000017152878750820312500061750363502953125000016672598145797343750003429791618564025000000540192179923833937500064823061590860072500005834075543177406525000381866762826157518000017184004327177088310004758647352141347532006118260881324589684029718395534545380311040
Technical details
Node id
n-d8e81e9a78e7e2d2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          1;
   2:          5,            9;
   3:         50,          180,           162;
   4:        625,         3375,          6075,          3645;
   5:       8750,        63000,        170100,        204120,          91854;
   6:     131250,      1181250,       4252500,       7654500,        6889050,        2480058;
   7:    2062500,     22275000,     100237500,     240570000,      324769500,      233834040,       70150212;
   8:   33515625,    422296875,    2280403125,    6841209375,    12314176875,    13299311025,     7979586615,     2051893701;
   9:  558593750,   8043750000,   50675625000,  182432250000,   410472562500,   591080490000,   531972441000,   273585826800,   61556811030;
  10: 9496093750, 153836718750, 1107624375000, 4652022375000, 12560460412500, 22608828742500, 27130594491000, 20929315750200, 9418192087590, 1883638417518;
%F Formula
The row polynomials satisfy n*P(n,x) = (20*n-30+(36*n-54)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((20*n - 30 + (36*n - 54)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((20*n - 30 + (36*n - 54)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

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