Real-rooted atlas

A013622 → x ↦ x/(x+1) → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 8\,x + 8\,n\right)\,P_{n}(x) = \left(9 + 48\,x + 64\,x^{2} + 24\,n + 64\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111819
25717664297
3243136821125124235
494577762188822528409657233
534833780015552029184022528032768746691
6123931671849072002488320350208021626882621449502009
7430116940084681152169344003483648039223296201850882097152118694587
814652927527042220825699864576270950400445906944418381824184549376167772161461537825
949207510550088990973445329981441797562368390168576053508833284303355904166094438413421772817791787123
1016336893936600042200352026425958401065996288028760997888520224768006115295232043033559040147639500801073741824214573239881
1153734591437646321732104000123787699205813710848018761534668842182796902465399685120067268247552042077257728012992276070485899345922567805035499
121753755351585206469007023365542732800029709047808011162324828163001845547008578506928947278479622144007175279738880403941674188811338713661446871947673630528348755377
1356864187182390551226824307328239224347648144111052800061794819440641934802970214444598848126976752059007631369068756336640074622909284352381908491960329826885173248549755813888360919263322915
141833471456368788944102138708672100144080691266982817341443228087582720011535032962252830956847523430462438387377766493589565394124810157007097036807597968945315843564479258296328466239533875243980465111044246293593703513
Technical details
Node id
n-d0d6912056bea0a0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f7ea89bf2d0df11e

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:     11,        8;
  2:     57,      176,       64;
  3:    243,     1368,     2112,       512;
  4:    945,     7776,    21888,     22528,       4096;
  5:   3483,    37800,   155520,    291840,     225280,      32768;
  6:  12393,   167184,   907200,   2488320,    3502080,    2162688,     262144;
  7:  43011,   694008,  4681152,  16934400,   34836480,   39223296,   20185088,    2097152;
  8: 146529,  2752704, 22208256,  99864576,  270950400,  445906944,  418381824,  184549376,   16777216;
  9: 492075, 10550088, 99097344, 532998144, 1797562368, 3901685760, 5350883328, 4303355904, 1660944384, 134217728;
%F Formula
The row polynomials satisfy (8*n-5+8*x)*P(n,x) = (24*n+9+(64*n+48)*x+64*x^2)*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[((24*n + 9 + (64*n + 48)*x + 64*x^2)*P[n - 1, x])/(8*n - 5 + 8*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] = ((24*n + 9 + (64*n + 48)*x + 64*x^2)*v[i - 1])/(8*n - 5 + 8*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

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.