Real-rooted atlas

A013622 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1282048
278411204002304
32195247040336008000110592
4614656175616018816008960001600005308416
517210368614656008780800062720000224000003200000254803968
648189030420652441603687936000351232000018816000005376000006400000012230590464
7134929285126746464256014456709120017210368000012293120000052684800000125440000001280000000587068342272
83778019983362158868561920539717140480077102448640006884147200000393379840000014049280000002867200000002560000000028179280429056
910578455953408680043597004801942981705728003238302842880003469610188800002478292992000001180139520000003612672000000064512000000005120000000001352605460594688
1029619676669542421156911906816006800435970048000129532113715200001619151421440000013878440755200000826097664000000033718272000000009031680000000001433600000000001024000000000064925062108545024
11829350946747187265163288672993280232726030974976000498698637803520000712426625433600000712426625433600000508876161024000000259630694400000000927252480000000002207744000000000031539200000000002048000000000003116402981210161152
1223221826508921241619904422721932492807819594640759193600186180824779980800002992191826821120000034196478020812800000284970650173440000001744718266368000000077889208320000000002472673280000000000529858560000000000688128000000000004096000000000000149587343098087735296
13650211142249794764860376748923195228160258757495385122406400677698202199130112000121017536106987520000015559397499469824000001481847380901888000000105846241492992000000056703343656960000000022501326848000000000064289505280000000000125239296000000000001490944000000000000819200000000000007180192468708211294208
14182059119829942534144182059119829942534144084527448492473319424002415069956927809126400047438874153939107840000677698202199130112000007261052166419251200000059273895236075520000000370461845225472000000001764104024883200000000063003715174400000000001636460134400000000000292225024000000000000321126400000000000001638400000000000000344649238497994142121984
Technical details
Node id
n-eeb76685a977fceb · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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:             28,             20;
  2:            784,           1120,             400;
  3:          21952,          47040,           33600,            8000;
  4:         614656,        1756160,         1881600,          896000,          160000;
  5:       17210368,       61465600,        87808000,        62720000,        22400000,         3200000;
  6:      481890304,     2065244160,      3687936000,      3512320000,      1881600000,       537600000,        64000000;
  7:    13492928512,    67464642560,    144567091200,    172103680000,    122931200000,     52684800000,     12544000000,     1280000000;
  8:   377801998336,  2158868561920,   5397171404800,   7710244864000,   6884147200000,   3933798400000,   1404928000000,   286720000000,   25600000000;
  9: 10578455953408, 68004359700480, 194298170572800, 323830284288000, 346961018880000, 247829299200000, 118013952000000, 36126720000000, 6451200000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (28+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[(28 + 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] = (28 + 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

Same fingerprint ?

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.