Real-rooted atlas

A034870 → Veronese 2 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
288117
3324818199
4128256160321577
5512128011204005013363
6204861446912358484072119601
7819228672394242688094081568981114243
832768131072212992180224844802150426881281665857
91310725898241105920111820865894422809644352432016213880899
1052428826214405570560655360046592002050048549120844806600200122619537
11209715211534336273940483676569630638080164003845637632120806415100896802421131836323
1283886085033164813212057619922944019051315212032409650692096140574722471040256256137282881768398401
1333554432218103808627048448104962457611331174408255569924127784961412136963236147247590404164161892833814478554083
1413421772893952409629360128005402263552649959833653692334083111714816127008768036118118469701632871270465228825480392126102926097

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Technical details
Node id
n-79d0a909b2a6e62d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-4006d85c148fe7de)
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:      2,      1;
  2:      8,      8,       1;
  3:     32,     48,      18,       1;
  4:    128,    256,     160,      32,      1;
  5:    512,   1280,    1120,     400,     50,      1;
  6:   2048,   6144,    6912,    3584,    840,     72,     1;
  7:   8192,  28672,   39424,   26880,   9408,   1568,    98,    1;
  8:  32768, 131072,  212992,  180224,  84480,  21504,  2688,  128,   1;
  9: 131072, 589824, 1105920, 1118208, 658944, 228096, 44352, 4320, 162, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 3)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 3)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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.