Real-rooted atlas

A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

A007318 → (θ+1)²

A125092 nonnegative coefficients

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

Recurrence from OEIS A125092
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 8\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
218918
3112271656
4116546425160
51209016012536432
6124135320375216491120
7128189560875756343642816

Show all 15 stored rows

Generated b-files: A001793 (row sums), A125092 ?

Computed in your browser ?

Other recurrences

Image under (θ+1)² · proved equal for all n (proof step)

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3} + 3\,n x + 5\,n x^{2} + 2\,n x^{3} + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-e7d0ffa461d9dbc4 · revision 2 · JSON
Definition source
oeis_catalog:A125092
derived:theta_plus_one_squared (equal by e-368ed38f1d3fa427)
Fingerprint
f6-d42e6339eab21b9c

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 0)
  • 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

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: 1;
  1: 1,  4;
  2: 1,  8,   9;
  3: 1, 12,  27,   16;
  4: 1, 16,  54,   64,   25;
  5: 1, 20,  90,  160,  125,   36;
  6: 1, 24, 135,  320,  375,  216,   49;
  7: 1, 28, 189,  560,  875,  756,  343,   64;
  8: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;
  9: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;
%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) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 8*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 8*x + 9*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] = 1; v[2] = 1 + 4*x; v[3] = 1 + 8*x + 9*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.

A125092: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,   9;
  3: 1, 12,  27,   16;
  4: 1, 16,  54,   64,   25;
  5: 1, 20,  90,  160,  125,   36;
  6: 1, 24, 135,  320,  375,  216,   49;
  7: 1, 28, 189,  560,  875,  756,  343,   64;
  8: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;
  9: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;
%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) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 8*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 8*x + 9*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[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] = 1 + 4*x; v[3] = 1 + 8*x + 9*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 = 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.

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-6f35c326bbf0f62d, e-7889e4573c43d5fc, e-85c5d5448d617672
  • 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-6f35c326bbf0f62d, e-7889e4573c43d5fc, e-85c5d5448d617672
  • 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-6f35c326bbf0f62d, e-7889e4573c43d5fc, e-85c5d5448d617672
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-6f35c326bbf0f62d, e-7889e4573c43d5fc

Proof chains

4 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 7 proof steps

Same fingerprint (12)

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.