Real-rooted atlas

A147716 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 32\,x + 17\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151732
22255102891024
333751147513005491332768
450625229500390150294780835211048576
575937543031259753750110542506264075141985733554432
61139062577456250219459375331627500281883375127787130241375691073741824
717085937513554843754608646875870522187598659181256708824325253444474541033867334359738368
8256289062523236875000921729375002089253250002959775437502683529730001520666847004924064076069757574411099511627776
93844335937539212226562517776209375004700819812500799139368125090569128387506843000811500332374325130094172725453511858787649735184372088832
10576650390625653537109375033330392578125100731853125000199784842031250271707385162500256612530431250166187162565000706295440901251778818147455020159939004491125899906842624
11864975585937510783362304687561105719726562520775944707031254709214133593750747195309196875084682135042312506855220455806250388462492495687514675249716503753326389935740853427189630763336028797018963968
1212974633789062517645501953125001099902955078125041551889414062500105957318005859375192135936650625000254046405126937500246787936409025000174808121623059375880514982990225002993750942166765061689413353739405826222372297611152921504606846976
131946195068359375286739406738281251949827965820312508102618435742187502295741890126953125468331345585898437570770069999646875008020607933293312500681751674329931562542925105420773468751945938112408397250601471780198959150113611336259803395990457803290593736893488147419103232
142919292602539062546319442626953125034121989401855468751546863519550781250048210579692666015625109277313970042968750185771433749073046875240618237998799375000238613086015476046875180285442767248568750102161750901440855625421030246139271405001192919030727935647520799613869102467701683778265594009291180591620717411303424
Technical details
Node id
n-507ad6d5bc6bc604 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-08a7d3764c1b9874

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,           17;
  2:         225,          510,           289;
  3:        3375,        11475,         13005,          4913;
  4:       50625,       229500,        390150,        294780,         83521;
  5:      759375,      4303125,       9753750,      11054250,       6264075,       1419857;
  6:    11390625,     77456250,     219459375,     331627500,     281883375,     127787130,      24137569;
  7:   170859375,   1355484375,    4608646875,    8705221875,    9865918125,    6708824325,    2534444745,     410338673;
  8:  2562890625,  23236875000,   92172937500,  208925325000,  295977543750,  268352973000,  152066684700,   49240640760,   6975757441;
  9: 38443359375, 392122265625, 1777620937500, 4700819812500, 7991393681250, 9056912838750, 6843000811500, 3323743251300, 941727254535, 118587876497;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+32*x+17*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[((15 + 32*x + 17*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] = ((15 + 32*x + 17*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.