Real-rooted atlas

A034870 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
28412
33232670
4128192808408
55121024672160102378
620485120460817922801213860
7819224576281601536040324481480782
83276811468815974411264042240806467216470832
913107252428886016074547236608010137614784960182744210
10524288235929644564484587520279552010250242196482534413202015994428
11209715210485760224133122673868819496960894566425625604392964118417602293222358
128388608461373441101004801494220801270087687018905625346048585728082368064064228824543339720
133355443220132659253057945680740352078446592050803507222226534465175552124467201464320960962912263166815962
141342177288724152322516582400424463564846425702403451650048177812275263504384015479193624893440248934413977636402818457556052

Other recurrences

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

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
Technical details
Node id
n-5ed189576f4a96f4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-8942b27fb46a4f4a)
Fingerprint
f6-9d7293d396b320fd

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:      2;
  2:      8,      4;
  3:     32,     32,      6;
  4:    128,    192,     80,      8;
  5:    512,   1024,    672,    160,     10;
  6:   2048,   5120,   4608,   1792,    280,     12;
  7:   8192,  24576,  28160,  15360,   4032,    448,    14;
  8:  32768, 114688, 159744, 112640,  42240,   8064,   672,  16;
  9: 131072, 524288, 860160, 745472, 366080, 101376, 14784, 960, 18;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(3*n-4)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (3*n - 4)*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (3*n - 4)*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.