Real-rooted atlas

A178821 → Veronese 2 (j=1) → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-8 - 8\,x + 6\,n + 2\,n x + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
155
23030
314035175
4560280840
5201615121263654
667206720126014700
72112026400792033055770
86336095040396003960201960
918304032032017160028600715704275
105125121025024672672160160100102380378
111397760314496024460807644008190013657836465
1237273609318400838656032614405096002184025225200
1397484802680832027417600127948802665600199920238079637180
142506752075202560861696004700160012337920137088042840247192920

Other recurrences

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

\(\left(-\frac{1}{2}\,n + \frac{1}{2}\,n^{2}\right)\,P_{n}(x) = \left(6\,x + \frac{7}{2}\,n x + \frac{1}{2}\,n^{2} x\right)\,P_{n-2}(x) + \left(-4 + 3\,n + n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)
Technical details
Node id
n-46bae8d0b3f1be4f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-739436df53dd23a9)
Fingerprint
f6-6cf8c5bbd0a3960a

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -1)
  • 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:      5;
  2:     30;
  3:    140,     35;
  4:    560,    280;
  5:   2016,   1512,    126;
  6:   6720,   6720,   1260;
  7:  21120,  26400,   7920,   330;
  8:  63360,  95040,  39600,  3960;
  9: 183040, 320320, 171600, 28600, 715;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (2*n^2+6*n-8+(n^2+2*n-8)*x)*P(n-1,x) - ((2*n+8)*x+(2*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 6*n - 8 + (n^2 + 2*n - 8)*x)*P[n - 1, x] - ((2*n + 8)*x + (2*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
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] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 6*n - 8 + (n^2 + 2*n - 8)*x)*v[i - 1] - ((2*n + 8)*x + (2*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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. [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.