Real-rooted atlas

A038271 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1171532
22895102251024
349131300511475337532768
483521294780390150229500506251048576
514198576264075110542509753750430312575937533554432
62413756912778713028188337533162750021945937577456250113906251073741824
741033867325344447456708824325986591812587052218754608646875135548437517085937534359738368
8697575744149240640760152066684700268352973000295977543750208925325000921729375002323687500025628906251099511627776
91185878764979417272545353323743251300684300081150090569128387507991393681250470081981250017776209375003921222656253844335937535184372088832
10201599390044917788181474550706295440901251661871625650002566125304312502717073851625001997848420312501007318531250003333039257812565353710937505766503906251125899906842624
11342718963076333326389935740851467524971650375388462492495687568552204558062508468213504231250747195309196875047092141335937502077594470703125611057197265625107833623046875864975585937536028797018963968
1258262223722976161689413353739402993750942166765088051498299022500174808121623059375246787936409025000254046405126937500192135936650625000105957318005859375415518894140625001099902955078125017645501953125001297463378906251152921504606846976
139904578032905937113611336259803395601471780198959150194593811240839725042925105420773468756817516743299315625802060793329331250070770069999646875004683313455858984375229574189012695312581026184357421875019498279658203125028673940673828125194619506835937536893488147419103232
141683778265594009292079961386910246770119291903072793564754210302461392714050010216175090144085562518028544276724856875023861308601547604687524061823799879937500018577143374907304687510927731397004296875048210579692666015625154686351955078125003412198940185546875463194426269531250291929260253906251180591620717411303424
Technical details
Node id
n-eed3e86f4ee981f1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
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:           17,           15;
  2:          289,          510,           225;
  3:         4913,        13005,         11475,          3375;
  4:        83521,       294780,        390150,        229500,         50625;
  5:      1419857,      6264075,      11054250,       9753750,       4303125,        759375;
  6:     24137569,    127787130,     281883375,     331627500,     219459375,      77456250,      11390625;
  7:    410338673,   2534444745,    6708824325,    9865918125,    8705221875,    4608646875,    1355484375,     170859375;
  8:   6975757441,  49240640760,  152066684700,  268352973000,  295977543750,  208925325000,   92172937500,   23236875000,   2562890625;
  9: 118587876497, 941727254535, 3323743251300, 6843000811500, 9056912838750, 7991393681250, 4700819812500, 1777620937500, 392122265625, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (17+15*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[(17 + 15*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] = (17 + 15*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.