Real-rooted atlas

A038324 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(21 + 10\,n x\right)\,P_{n}(x) = \left(441 + 420\,x + 100\,x^{2} + 210\,n x + 100\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1212041
24418403001581
392612646018900400058621
4194481740880793800336000500002115161
54084101194481002778300017640000525000060000074805201
6857661214900921208751645007408800003307500007560000070000002605252741
7180108854112007256940257298363002722734000016206750000555660000010290000008000000089637871781

All 15 rows

Technical details
Node id
n-aac1596390676dab · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-929a675042542dd5

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:           21,            20;
  2:          441,           840,            300;
  3:         9261,         26460,          18900,           4000;
  4:       194481,        740880,         793800,         336000,          50000;
  5:      4084101,      19448100,       27783000,       17640000,        5250000,         600000;
  6:     85766121,     490092120,      875164500,      740880000,      330750000,       75600000,       7000000;
  7:   1801088541,   12007256940,    25729836300,    27227340000,    16206750000,     5556600000,    1029000000,      80000000;
  8:  37822859361,  288174166560,   720435416400,   914838624000,   680683500000,   311169600000,   86436000000,   13440000000,    900000000;
  9: 794280046581, 6808114684980, 19451756242800, 28817416656000, 25729836300000, 14702763600000, 5445468000000, 1270080000000, 170100000000, 10000000000;
%F Formula
The row polynomials satisfy (21+10*n*x)*P(n,x) = (441+(210*n+420)*x+(100*n+100)*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[((441 + (210*n + 420)*x + (100*n + 100)*x^2)*P[n - 1, x])/(21 + 10*n*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] = ((441 + (210*n + 420)*x + (100*n + 100)*x^2)*v[i - 1])/(21 + 10*n*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

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.