Real-rooted atlas

A038220 → Veronese 2 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-13 - 4\,x + 13\,n + 4\,n x\right)\,P_{n-1}(x) + \left(10 + 2\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
213417
3633699
431324816577
5156315602403363
67813937223526419601
73906354684191521344114243
819531331249614064017152256665857
9976563175780896681617280069123880899
104882813976562063476801513600108800102422619537
1124414063537109324028323212084864130944033792131836323
1212207031329296874424902347290331904133670406328324096768398401
136103515631586914056150756840064270003212199449688657921597444478554083
143051757813854492186889721680164400634560102645939210349875234693121638426102926097

Other recurrences

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

\(P_{n}(x) = \left(-5 + 4\,x\right)\,P_{n-2}(x) + \left(6\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)
Technical details
Node id
n-9cb23cc1a7a54971 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-dfa50262c79d2eb1)
Fingerprint
f6-dcf9dad318e02ab9

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      3;
  2:     13,       4;
  3:     63,      36;
  4:    313,     248,     16;
  5:   1563,    1560,    240;
  6:   7813,    9372,   2352,     64;
  7:  39063,   54684,  19152,   1344;
  8: 195313,  312496, 140640,  17152,  256;
  9: 976563, 1757808, 966816, 172800, 6912;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (13*n-13+(4*n-4)*x)*P(n-1,x) + (10+2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 13 + (4*n - 4)*x)*P[n - 1, x] + (10 + 2*x - 8*x^2)*D[P[n - 1, x], x])/(3*n - 3)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((13*n - 13 + (4*n - 4)*x)*v[i - 1] + (10 + 2*x - 8*x^2)*deriv(v[i - 1], x))/(3*n - 3)); 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.