Real-rooted atlas

A098172: Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

Veronese parent of A098172 → Veronese 3 (j=1)

A098172 ↗ nonnegative coefficients

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

Recurrence from A098172

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2011
30011
400112
5000415
6000010111
70000120122
800000735143
90000002856185
10000000184841170
110000000102101201341
1200000000554621651683
1300000000122092422011366
1400000000013715171628612731
150000000000912002300336415461

b-files: A024493 (row sums), A098172, A130781 (row sums) ?

Technical details
Node id
n-e94410689654e39c · revision 0 · JSON
Definition source
oeis_catalog:A098172
Fingerprint
f6-be512ecc8991b097

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 ?

  • A098172 triangle 91 terms, 13 rows · tabl · nonn · catalog b-file?

Row sums ? A024493, A130781

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 0, 1;
   4: 0, 0, 1, 1;
   5: 0, 0, 0, 4,  1;
   6: 0, 0, 0, 0, 10,  1;
   7: 0, 0, 0, 0,  1, 20,  1;
   8: 0, 0, 0, 0,  0,  7, 35,  1;
   9: 0, 0, 0, 0,  0,  0, 28, 56,  1;
  10: 0, 0, 0, 0,  0,  0,  1, 84, 84, 1;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 3*x^2*P(n-2,x) + (x^2+x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = x, P(3,x) = x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[3, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 3*x^2*P[n - 2, x] + (x^2 + x^3)*P[n - 3, 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] = x; v[3] = x^2; for(n = 4, N, my(i = n); v[i] = 3*x*v[i - 1] - 3*x^2*v[i - 2] + (x^2 + x^3)*v[i - 3]); 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

A098172 triangle, offset 0, from P1
%e Example
Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 0, 1;
  3: 0, 0, 1, 1;
  4: 0, 0, 0, 4,  1;
  5: 0, 0, 0, 0, 10,  1;
  6: 0, 0, 0, 0,  1, 20,  1;
  7: 0, 0, 0, 0,  0,  7, 35,  1;
  8: 0, 0, 0, 0,  0,  0, 28, 56,  1;
  9: 0, 0, 0, 0,  0,  0,  1, 84, 84, 1;
%F Formula
The row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 3*x^2*P(n-2,x) + (x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 3*x^2*P[n - 2, x] + (x^2 + x^3)*P[n - 3, 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] = x; v[3] = x^2; for(n = 3, N, my(i = n + 1); v[i] = 3*x*v[i - 1] - 3*x^2*v[i - 2] + (x^2 + x^3)*v[i - 3]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.