Real-rooted atlas

A038255 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1279
24284981
3884294343729
4162241176274424016561
532560392013720240101680759049
66413441176054880144060201684117649531441
7128313632928192080672280141178816470868235434782969
8256716887808614656268912075295361317668813176688576480143046721
95121612822579218439689680832338829127906012811859019210376641840353607387420489
1010243584056448052684803226944013553164839530064079060128010376641808070721402824752493486784401
1120487884813798401448832010141824049694937617393228164348307040760953732088777935406214455478197732674331381059609
12409617203233116163863552030425472017038264326957291264208718737924565722392071022348320745734657364745584183213841287201282429536481
138192372736782745610045235287895808055374359042584136755290444786432237417564384461645264080646303369712616925943816359873467226968890104072541865828329
1416384802816182640642556968962461082624172275783689044478643236177914572811079486337922585213478848452412358798457579754756165038228541164271289229139667822307284922876792454961
Technical details
Node id
n-2a3c47627a03f4f8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8d10fbbacee008e8

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:   2,     7;
  2:   4,    28,     49;
  3:   8,    84,    294,     343;
  4:  16,   224,   1176,    2744,    2401;
  5:  32,   560,   3920,   13720,   24010,    16807;
  6:  64,  1344,  11760,   54880,  144060,   201684,   117649;
  7: 128,  3136,  32928,  192080,  672280,  1411788,  1647086,    823543;
  8: 256,  7168,  87808,  614656, 2689120,  7529536, 13176688,  13176688,   5764801;
  9: 512, 16128, 225792, 1843968, 9680832, 33882912, 79060128, 118590192, 103766418, 40353607;
%F Formula
The row polynomials satisfy P(n,x) = (2+7*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[(2 + 7*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] = (2 + 7*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.