Real-rooted atlas

A013609 → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
192433
2814325761089
37295832155521382435937
46561699842799364976643317761185921
55904978732041990401119744014929920796262439135393
65314418503056566870402015539204031078404299816961911029761291467969
7478296989282088714256704317447424084652646401354442342412039487488458647142442618442977
8430467219183300488571080448457124290561523747635203250661621764334215495683302259425281100753141761406408618241
93874204899298091736991796451846171177922562468471169024658258978406411702381838336133741506723848916100448256264180754022446411484401953
103486784401929809173601115771008320793437161472037027067535360118486616113152263303591362560401224520171520401224520171520237762678620160634033809653761531578985264449
1131381059609920511081864122734810915209818784873216052366852657152019550291658670085213411108978688993030687424512013240409165660160117692525916979206276934715572224152168114316902450542106513726817
122824295364819037745167392132553595788416117825418478592070695251087155203016330713051955293841399961616384214494628483694592357491047472824320423693093301125120338954474640900096164341563462254592365203474360565761667889514952984961
132541865828329881180153820721409888246113152137855739619952649190382641330176044113836678384844815684919707870167044182645255432044544836529051086408908812393022979057909760132192245109951037449613981462541893632427288065001861939287648833846535782455040353993448503713
142287679245496185406691831854414803826584188096157907483564673024115798821280760217661759371349738782722470374853989551308875287614597776801792175671100728145870848312304179072259325952416405572096345767936403787221426759532544269191480951173021696110437530646635085824210357201231685877761816331681783800622529
Technical details
Node id
n-9f4dd10196cdd660 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-10cba46fcb583bff

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:         9,         24;
  2:        81,        432,         576;
  3:       729,       5832,       15552,        13824;
  4:      6561,      69984,      279936,       497664,        331776;
  5:     59049,     787320,     4199040,     11197440,      14929920,       7962624;
  6:    531441,    8503056,    56687040,    201553920,     403107840,     429981696,      191102976;
  7:   4782969,   89282088,   714256704,   3174474240,    8465264640,   13544423424,    12039487488,     4586471424;
  8:  43046721,  918330048,  8571080448,  45712429056,  152374763520,  325066162176,   433421549568,   330225942528,  110075314176;
  9: 387420489, 9298091736, 99179645184, 617117792256, 2468471169024, 6582589784064, 11702381838336, 13374150672384, 8916100448256, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (9+24*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[(9 + 24*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] = (9 + 24*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.