Real-rooted atlas

A038221 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(12 + 36\,x + 27\,x^{2} + 18\,n + 27\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115924
214427081495
31188388836457299450
490724276869984437406561172125
5660964082409622801049760492075590493037500
64665603569184110224801732104014171760531441053144152396875
7321926429393280112429296231472080272806380178564176558013054782969888468750
821835008231787008105815808026983031044166497440392841187221427701125739562804304672114864765625
91461265921768635648938737382428570268160546406378566749725852853033560272247949112965811307335387420489246037500000
10967458816131513932807958860416028162121472064283103360098353148140810124588779206818600606402789427520805811307335034867844014036552734375
1163489484809577842278465099396736026264239372806970125064320127280544652801622826944323214319061273440843801825042030683702728805753194261653138105960965738144531250
1241358864384685686435840517203483033623435782824960709134463065601505547013893122291049803750402503790142670081933073271914401012562190050403313839894710456485907296202824295364811063919970703125
1326774422732848389871329284011265649664020170935838310468549664763008016593746435735042935816677091584382932610055424036617930836549922512995253488720118469776235896835247206152828855073759613795254186582832917126516601562500
141724011610112337357726433283048561893744641684731572858880635384478906777617274515520278016348468675150435845284470018764851260311886083729280512651031711698883166374019395787213570174368839088370095664604702453379182394909022876792454961274413504638671875
Technical details
Node id
n-f0e87864363473c6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:        15,          9;
  2:       144,        270,         81;
  3:      1188,       3888,       3645,         729;
  4:      9072,      42768,      69984,       43740,        6561;
  5:     66096,     408240,     962280,     1049760,      492075,       59049;
  6:    466560,    3569184,   11022480,    17321040,    14171760,     5314410,      531441;
  7:   3219264,   29393280,  112429296,   231472080,   272806380,   178564176,    55801305,     4782969;
  8:  21835008,  231787008, 1058158080,  2698303104,  4166497440,  3928411872,  2142770112,   573956280,   43046721;
  9: 146126592, 1768635648, 9387373824, 28570268160, 54640637856, 67497258528, 53033560272, 24794911296, 5811307335, 387420489;
%F Formula
The row polynomials satisfy (3*n-1+3*x)*P(n,x) = (18*n+12+(27*n+36)*x+27*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[((18*n + 12 + (27*n + 36)*x + 27*x^2)*P[n - 1, x])/(3*n - 1 + 3*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] = ((18*n + 12 + (27*n + 36)*x + 27*x^2)*v[i - 1])/(3*n - 1 + 3*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.