Real-rooted atlas

A038255 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 17\,x + 10\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171017
249140100289
33431470210010004913
424011372029400280001000083521
5168071200503430004900003500001000001419857
611764910084203601500686000073500004200000100000024137569
7823543823543035294700840350001200500001029000004900000010000000410338673
85764801658834403294172009411920001680700000192080000013720000005600000001000000006975757441
940353607518832090296475480098825160002117682000030252600000288120000001764000000063000000001000000000118587876497
102824752494035360700259416045009882516000024706290000042353640000050421000000041160000000022050000000070000000000100000000002015993900449
1119773267433107227739022194483850095119216500027176919000005435383800000776483400000079233000000005659500000000269500000000077000000000010000000000034271896307633
1213841287201237279209160186433664340088777935400002853576495000065224605600000108707676000000133111440000000118849500000000754600000000003234000000000084000000000001000000000000582622237229761
139688901040717993673361301542314859540080787921214000288528290050000741929888700000141319978800000020188568400000002163060900000000171671500000000098098000000000038220000000000091000000000000100000000000009904578032905937
1467822307284913564461456980125955713529100719746934452000282757724249000080787921214000001731169740300000028263995760000000353299947000000003364761400000000024034010000000000124852000000000004459000000000000980000000000000100000000000000168377826559400929
Technical details
Node id
n-1282f5faa3eb349f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0fbe6fa5f58efdcf

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:        7,        10;
  2:       49,       140,        100;
  3:      343,      1470,       2100,       1000;
  4:     2401,     13720,      29400,      28000,       10000;
  5:    16807,    120050,     343000,     490000,      350000,      100000;
  6:   117649,   1008420,    3601500,    6860000,     7350000,     4200000,     1000000;
  7:   823543,   8235430,   35294700,   84035000,   120050000,   102900000,    49000000,    10000000;
  8:  5764801,  65883440,  329417200,  941192000,  1680700000,  1920800000,  1372000000,   560000000,  100000000;
  9: 40353607, 518832090, 2964754800, 9882516000, 21176820000, 30252600000, 28812000000, 17640000000, 6300000000, 1000000000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7+17*x+10*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[((7 + 17*x + 10*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] = ((7 + 17*x + 10*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.