Real-rooted atlas

A235706: (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).

A132440 → D+1 → (1/n)D²+(2/n)D+1/n

A235706 ↗ nonnegative coefficients

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

Recurrence from A235706

\(P_{n}(x) = \left(5 + x - n\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
366113
46189134
50243612173
6006060151136
700012090181229
80000210126211358

All 15 rows b-files: A135859 (row sums), A235706 ?

Technical details
Node id
n-77dad805f3db7972 · revision 0 · JSON
Definition source
oeis_catalog:A235706
Fingerprint
f6-607f8746089a3f6e

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 -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 ?

  • A235706 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A135859

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 3,  1;
   3: 6,  6,  1;
   4: 6, 18,  9,   1;
   5: 0, 24, 36,  12,   1;
   6: 0,  0, 60,  60,  15,   1;
   7: 0,  0,  0, 120,  90,  18,   1;
   8: 0,  0,  0,   0, 210, 126,  21,   1;
   9: 0,  0,  0,   0,   0, 336, 168,  24,  1;
  10: 0,  0,  0,   0,   0,   0, 504, 216, 27, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-5-x)*P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 5 - x)*P[n - 1, x] + (n - 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(n - 5 - x)*v[i - 1] + (n - 2)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A235706 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 3,  1;
  2: 6,  6,  1;
  3: 6, 18,  9,   1;
  4: 0, 24, 36,  12,   1;
  5: 0,  0, 60,  60,  15,   1;
  6: 0,  0,  0, 120,  90,  18,   1;
  7: 0,  0,  0,   0, 210, 126,  21,   1;
  8: 0,  0,  0,   0,   0, 336, 168,  24,  1;
  9: 0,  0,  0,   0,   0,   0, 504, 216, 27, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-4-x)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 4 - x)*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n - 4 - x)*v[i - 1] + (n - 1)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.