Real-rooted atlas

A034870 → Veronese 2 (j=1) → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(12\,x - n - 19\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
244
3201232
45611224192
5120504360401024
622015842376880605120
7364400410296800818208424576
856087363432045760218403360112114688
98161713695472194480159120514085712144524288
1011403100823256067184083980046512010852891201802359296
1115405266851163219896803527160298452011938082106721386022010485760
12202485008103831252300161248072014976864915252827688323825362024026446137344
13260013156019734001249820038630800624036005408312024996400592020065780028600312201326592
1432761965603552120276276001073709002246529602620951201717934406216210011840400108108039312364872415232
Technical details
Node id
n-56ada67cf290526a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-7cd476ceab5c0aa5

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:   0;
  1:   0;
  2:   4;
  3:  20,    12;
  4:  56,   112,    24;
  5: 120,   504,   360,     40;
  6: 220,  1584,  2376,    880,     60;
  7: 364,  4004, 10296,   8008,   1820,    84;
  8: 560,  8736, 34320,  45760,  21840,  3360,  112;
  9: 816, 17136, 95472, 194480, 159120, 51408, 5712, 144;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2-n+(6*n^2-19*n+12)*x)*P(n-1,x) + ((4*n-4)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (6*n^2 - 19*n + 12)*x)*P[n - 1, x] + ((4*n - 4)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - n + (6*n^2 - 19*n + 12)*x)*v[i - 1] + ((4*n - 4)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.