Real-rooted atlas

A038324 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(41 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1412061
2168116404003721
368921100860492008000226981
4282576155136804034400131200016000013845841
5115856201282576100275684000134480000328000003200000844596301
6475010424113902744120169545660001102736000040344000007872000006400000051520374361
71947542738816650145937409731920884007912130800003859576000001129632000001836800000012800000003142742836021
879849252291213116068382096053201167499200519035780480003164852320000012350643200000301235200000041984000000025600000000191707312997281
93273819343939611437286541241780280446154388640031920700499520002335661012160000113934683520000037051929600000077460480000000944640000000051200000000011694146092834141
101342265931015240165476386878792200143728654124178000186964102925760000159603502497600000934264404864000003797822784000000010586265600000000193651200000000020992000000000010240000000000713342911662882601
1155032903171624844129529850482335282207202402556667142000105401013024397200001028302566091680000070225541098944000003425636151168000000119360144640000000029112230400000000047336960000000000461824000000000020480000000000043513917611435838661
1222563490300366186081132078967611899625840354358205788023386400576192204533371360000632406078146383200000493585231724006400000280902164395776000000117450382325760000000358080433920000000007763261440000000000113608704000000000010076160000000000040960000000000002654348974297586158321
1392510310231501362932158665074780952083810601717026578954695135920030711044501628693488000374524932946691384000003288511606361192640000021388693374706944000000104335089632716800000003817137425587200000000103445458688000000000020184479744000000000026852966400000000000218316800000000000081920000000000000161915287432152755657581
143792922719491555880216125902886864820381620988082131104693332917334840016025581403577154601920002149773115114008544160000209733962450147175040000015346387496352232320000008555477349882777600000003651728137145088000000001187553865738240000000002896472843264000000000051378675712000000000006265692160000000000004702208000000000000016384000000000000009876832533361318095112441
Technical details
Node id
n-89e4696a36e5da90 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-8f1c79a7a65ded7e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:              41,               20;
  2:            1681,             1640,              400;
  3:           68921,           100860,            49200,             8000;
  4:         2825761,          5513680,          4034400,          1312000,           160000;
  5:       115856201,        282576100,        275684000,        134480000,         32800000,          3200000;
  6:      4750104241,      13902744120,      16954566000,      11027360000,       4034400000,        787200000,        64000000;
  7:    194754273881,     665014593740,     973192088400,     791213080000,     385957600000,     112963200000,     18368000000,     1280000000;
  8:   7984925229121,   31160683820960,   53201167499200,   51903578048000,   31648523200000,   12350643200000,   3012352000000,   419840000000,   25600000000;
  9: 327381934393961, 1437286541241780, 2804461543886400, 3192070049952000, 2335661012160000, 1139346835200000, 370519296000000, 77460480000000, 9446400000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (41+20*x)*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[(41 + 20*x)*P[n - 1, 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] = (41 + 20*x)*v[i - 1]); 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

Proof chains ? JSON

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.