Real-rooted atlas

A034870 → Veronese 2 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-1 - 4\,x - 5\,x^{2} - 2\,x^{3} + n + 4\,n x + 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
218817
3118483299
4132160256128577
5150400112012805123363
6172840358469126144204819601
7198156894082688039424286728192114243
811282688215048448018022421299213107232768665857
91162432044352228096658944111820811059205898241310723880899
1012006600844805491202050048465920065536005570560262144052428822619537
11124296801510081208064563763216400384306380803676569627394048115343362097152131836323
1212881372825625624710401405747250692096120324096190513152199229440132120576503316488388608768398401
1313381892841641647590403236147214121369641277849682555699211331174401049624576627048448218103808335544324478554083
1413922548065228887127046970163236118118412700876803111714816536923340864995983365402263552293601280093952409613421772826102926097
Technical details
Node id
n-ffd5c5651e6f274b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-607690c148a1ebbd

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,   2;
  2: 1,   8,    8;
  3: 1,  18,   48,    32;
  4: 1,  32,  160,   256,    128;
  5: 1,  50,  400,  1120,   1280,    512;
  6: 1,  72,  840,  3584,   6912,   6144,    2048;
  7: 1,  98, 1568,  9408,  26880,  39424,   28672,    8192;
  8: 1, 128, 2688, 21504,  84480, 180224,  212992,  131072,  32768;
  9: 1, 162, 4320, 44352, 228096, 658944, 1118208, 1105920, 589824, 131072;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n-1+(4*n-4)*x+(5*n-5)*x^2+(2*n-2)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (4*n - 4)*x + (5*n - 5)*x^2 + (2*n - 2)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (4*n - 4)*x + (5*n - 5)*x^2 + (2*n - 2)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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.