Real-rooted atlas

A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

A007318 → (−1/2)D²+(−1/2+(1/2)n)D

A129533 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence from OEIS A129533
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
2011
30336
40612624
501030301080
601560906015240
702110521021010521672

Show all 15 stored rows

Generated b-files: A001788 (row sums), A094305, A094305 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-ccd988a5d83adc8b · revision 1 · JSON
Definition source
oeis_catalog:A129533
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: -1, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 0;
  1: 0;
  2: 0,  1;
  3: 0,  3,   3;
  4: 0,  6,  12,   6;
  5: 0, 10,  30,  30,   10;
  6: 0, 15,  60,  90,   60,   15;
  7: 0, 21, 105, 210,  210,  105,  21;
  8: 0, 28, 168, 420,  560,  420, 168,  28;
  9: 0, 36, 252, 756, 1260, 1260, 756, 252, 36;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
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] = 0; v[3] = x; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A094305: triangle, offset 0, from the row P2, low zeros dropped
%e Example
Triangle begins:
  0:  1;
  1:  3,   3;
  2:  6,  12,    6;
  3: 10,  30,   30,   10;
  4: 15,  60,   90,   60,   15;
  5: 21, 105,  210,  210,  105,   21;
  6: 28, 168,  420,  560,  420,  168,   28;
  7: 36, 252,  756, 1260, 1260,  756,  252,   36;
  8: 45, 360, 1260, 2520, 3150, 2520, 1260,  360,  45;
  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = x, P(1,x) = 3*x + 3*x^2, P(2,x) = 6*x + 12*x^2 + 6*x^3.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 3*x + 3*x^2;
P[2, x_] := 6*x + 12*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 3*x + 3*x^2; v[3] = 6*x + 12*x^2 + 6*x^3; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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)

Generated with polytool c48629374597.

A094305: reversed triangle, offset 0, from the row P2, low zeros dropped
%e Example
Triangle begins:
  0:  1;
  1:  3,   3;
  2:  6,  12,    6;
  3: 10,  30,   30,   10;
  4: 15,  60,   90,   60,   15;
  5: 21, 105,  210,  210,  105,   21;
  6: 28, 168,  420,  560,  420,  168,   28;
  7: 36, 252,  756, 1260, 1260,  756,  252,   36;
  8: 45, 360, 1260, 2520, 3150, 2520, 1260,  360,  45;
  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = x, P(1,x) = 3*x + 3*x^2, P(2,x) = 6*x + 12*x^2 + 6*x^3.
%t Mathematica
P[0, x_] := x;
P[1, x_] := 3*x + 3*x^2;
P[2, x_] := 6*x + 12*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 3*x + 3*x^2; v[3] = 6*x + 12*x^2 + 6*x^3; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-177701ed6604099d

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Same fingerprint (59)

Equal up to content, sign, reversal and a small index shift.

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. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.