Real-rooted atlas

A038326 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 34\,x + 23\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112334
21215065291156
313318349174571216739304
4146411224523840545353482798411336336
5161051168371570409901472207015391255643634345435424
61771561222250381161763353238855405079114154247986381480358891544804416
71948717128522132117891155596234796645130363929851635474756311398763453340482544752523350144
8214358881358563946426240361532109732420952286800645670479739261848501545591932299624639336783109852811785793904896
9235794769144372288367371113684524181058494570856786527842661187354673073816551004533756148314196471327752787542819180115266146360716992766464
1025937424601542327968930510281316220528452049146840104108634378210261218028076236455152624678290543818720394840426403314855045198126792760930414265112136492064377754059776
112853116706116562168424053686044880696454303372433459551799592108537630526789689953742611014693517214618164505162919439101719826703248681511986670962036265501260785685152995280975791392770188843638032384
1231384283767217874602108863690557924251931463116129024073402969326979087089599337484391277176242323257378721596434293630107319224567542812072064895527413522329595660330832118552200914125770888044638364219146244320203212386420683693101056
1334522712143931938390084639579117725301527510829025606450442496247179306445494866517756575334940795214950317972165312604103506648508911081721623172442526105599718855033423283044845157696643176549102348992618495191643026313379129377890590350403636193646738381138303245565435904
143797498335832411111631331034578215108080362697221912635849030619494687265613192606209441303834733508986940269529362096418226762722771462671960437978441661426024836710392358073502943711778122660713207622971404400946162108275317100866824130192962097575453177621599738215976982115928363245387498092758702310349224820736
Technical details
Node id
n-301ed9476340747f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2f74b0bddfe64ff1

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,          23;
  2:        121,         506,          529;
  3:       1331,        8349,        17457,         12167;
  4:      14641,      122452,       384054,        535348,        279841;
  5:     161051,     1683715,      7040990,      14722070,      15391255,        6436343;
  6:    1771561,    22225038,    116176335,     323885540,     507911415,      424798638,      148035889;
  7:   19487171,   285221321,   1789115559,    6234796645,   13036392985,    16354747563,    11398763453,     3404825447;
  8:  214358881,  3585639464,  26240361532,  109732420952,  286800645670,   479739261848,   501545591932,   299624639336,   78310985281;
  9: 2357947691, 44372288367, 371113684524, 1810584945708, 5678652784266, 11873546730738, 16551004533756, 14831419647132, 7752787542819, 1801152661463;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+34*x+23*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[((11 + 34*x + 23*x^2)*P[n - 1, x])/(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] = ((11 + 34*x + 23*x^2)*v[i - 1])/(1 + 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.