Real-rooted atlas

A013622 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(9 + 30\,x + 25\,x^{2} + 15\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18513
2398025144
31625856001251472
462132405850400062514336
52268155254050048750250003125135168
6801968040232875405000365625150000156251245184
72770228066511907002716875354375025593758750007812511272192

All 15 rows

Technical details
Node id
n-b99042e296c8e644 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-951a7e0f7b8ffcff

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:      8,       5;
  2:     39,      80,       25;
  3:    162,     585,      600,      125;
  4:    621,    3240,     5850,     4000,       625;
  5:   2268,   15525,    40500,    48750,     25000,      3125;
  6:   8019,   68040,   232875,   405000,    365625,    150000,     15625;
  7:  27702,  280665,  1190700,  2716875,   3543750,   2559375,    875000,     78125;
  8:  94041, 1108080,  5613300, 15876000,  27168750,  28350000,  17062500,   5000000,   390625;
  9: 314928, 4231845, 24931800, 84199500, 178605000, 244518750, 212625000, 109687500, 28125000, 1953125;
%F Formula
The row polynomials satisfy (5*n-2+5*x)*P(n,x) = (15*n+9+(25*n+30)*x+25*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[((15*n + 9 + (25*n + 30)*x + 25*x^2)*P[n - 1, x])/(5*n - 2 + 5*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] = ((15*n + 9 + (25*n + 30)*x + 25*x^2)*v[i - 1])/(5*n - 2 + 5*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.