Real-rooted atlas

A137597 → D+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 2\,x - x^{2} + n x + n x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2171018
3112403790
4118103221151494
512521579012416742946
6133395218058367218326318926
7142665511020496429384380717007130066
81521050106825959118635431998527811194828950654
91631578204961513476569011676473243709818472425625957353794
1017522803678034685719906537031515151235401906215112825759353502759954638
11188319062535732897536982625037881742974101379294511535071349296236523430840513333618
121102434510169514494921319907678496594306819414783200726127827348012742863887022945681632548854601380766
131117578515930227132823005874322208329011056822923708735602829477563212076730172109085469485521005933119205922343062556322
1411337553241696484784364211147577149001356698513615214459260445987542008868383760611654803050896293931189450931768519097183602419742815726
Technical details
Node id
n-bf87ef829751e42e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f80b58a897df2b0a

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  7,   10;
  3: 1, 12,   40,    37;
  4: 1, 18,  103,   221,    151;
  5: 1, 25,  215,   790,   1241,    674;
  6: 1, 33,  395,  2180,   5836,   7218,    3263;
  7: 1, 42,  665,  5110,  20496,  42938,   43807,   17007;
  8: 1, 52, 1050, 10682,  59591, 186354,  319985,  278111,   94828;
  9: 1, 63, 1578, 20496, 151347, 656901, 1676473, 2437098, 1847242, 562595;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+2)*x+(n-1)*x^2)*P(n-1,x) - (x^2+x^3)*d/dx 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[(1 + (n + 2)*x + (n - 1)*x^2)*P[n - 1, x] - (x^2 + x^3)*D[P[n - 1, x], 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] = (1 + (n + 2)*x + (n - 1)*x^2)*v[i - 1] - (x^2 + x^3)*deriv(v[i - 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

Candidate equalities ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.