Real-rooted atlas

A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

A007318 → ×(−(1/2)n+(1/2)n²)

A237765 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A237765
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1 + 2\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
00
10
21214
3399324
46243624696
510501001005010320
615902253002259015960
721147441735735441147212688
828224784156819601568784224287168
9363241296302445364536302412963243618432
1045450202554009450113409450540020254504546080
115560530259075181502541025410181509075302560555112640
1266792435614520326705227260984522723267014520435679266270336
137810146084223085577010038613384813384810038655770223086084101478638976
1491127482813312491091182182273273312312273273182182910913312482811274911490944

Generated b-files: A100381 (row sums) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1 + 2\,x + x^{2}\)
Technical details
Node id
n-a12ea22caf034cea · revision 2 · JSON
Definition source
oeis_catalog:A237765
derived:reversal (equal by e-a8e50b0e95b009dc)
Fingerprint
f6-77a7ed3b1d5f257e

Observed on the stored rows ?

  • Degrees: -1, -1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:  1,   2,    1;
  3:  3,   9,    9,    3;
  4:  6,  24,   36,   24,    6;
  5: 10,  50,  100,  100,   50,   10;
  6: 15,  90,  225,  300,  225,   90,   15;
  7: 21, 147,  441,  735,  735,  441,  147,   21;
  8: 28, 224,  784, 1568, 1960, 1568,  784,  224,  28;
  9: 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 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) = 1 + 2*x + x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1 + 2*x + x^2;
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] = 1 + 2*x + x^2; 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.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a72beb3d57d8ea71, e-ccfad80d3bf960c1
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-a72beb3d57d8ea71, e-ccfad80d3bf960c1
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-a72beb3d57d8ea71, e-ccfad80d3bf960c1

Proof chains

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

Used in 6 proof steps

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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.