Real-rooted atlas

A391045: Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.

A172431 → ⊙ C(n,k+1) (shift -1)

A391045 nonnegative coefficients

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

Recurrence from OEIS A391045
\(\left(n + 2\,n^{2}\right)\,P_{n}(x) = \left(4\,x + n - 23\,n x + 2\,n^{2} + 13\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 3\,x^{2} + 4\,n x - \frac{27}{2}\,n x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2} - 6\,n x^{2} + \frac{5}{2}\,n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31416
41910121
51164220180
6125120140351322
71362756003785611347
8149546192523108828415798

Show all 15 stored rows

Generated b-files: A391045 ?

Computed in your browser ?

Technical details
Node id
n-5c0c5bce3aee8449 · revision 1 · JSON
Definition source
oeis_catalog:A391045
Fingerprint
f6-996d7a7fb08a10c3

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

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  4,    1;
   4: 1,  9,   10,     1;
   5: 1, 16,   42,    20,     1;
   6: 1, 25,  120,   140,    35,     1;
   7: 1, 36,  275,   600,   378,    56,     1;
   8: 1, 49,  546,  1925,  2310,   882,    84,    1;
   9: 1, 64,  980,  5096, 10010,  7392,  1848,  120,   1;
  10: 1, 81, 1632, 11760, 34398, 42042, 20592, 3564, 165, 1;
%F Formula
The row polynomials satisfy (2*n^2+n)*P(n,x) = (2*n^2+n+(13*n^2-23*n+4)*x)*P(n-1,x) + ((4*n-1)*x-((27/2)*n-3)*x^2)*d/dx P(n-1,x) + ((5/2)*n^2-6*n+2)*x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (13*n^2 - 23*n + 4)*x)*P[n - 1, x] + ((4*n - 1)*x - ((27/2)*n - 3)*x^2)*D[P[n - 1, x], x] + ((5/2)*n^2 - 6*n + 2)*x^2*P[n - 2, x])/(2*n^2 + n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n^2 + n + (13*n^2 - 23*n + 4)*x)*v[i - 1] + ((4*n - 1)*x - ((27/2)*n - 3)*x^2)*deriv(v[i - 1], x) + ((5/2)*n^2 - 6*n + 2)*x^2*v[i - 2])/(2*n^2 + n)); 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)

Generated with polytool c48629374597.

A391045: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,   10,     1;
  4: 1, 16,   42,    20,     1;
  5: 1, 25,  120,   140,    35,     1;
  6: 1, 36,  275,   600,   378,    56,     1;
  7: 1, 49,  546,  1925,  2310,   882,    84,    1;
  8: 1, 64,  980,  5096, 10010,  7392,  1848,  120,   1;
  9: 1, 81, 1632, 11760, 34398, 42042, 20592, 3564, 165, 1;
%F Formula
The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (2*n^2+5*n+3+(13*n^2+3*n-6)*x)*P(n-1,x) + ((4*n+3)*x-((27/2)*n+21/2)*x^2)*d/dx P(n-1,x) + ((5/2)*n^2-n-3/2)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 5*n + 3 + (13*n^2 + 3*n - 6)*x)*P[n - 1, x] + ((4*n + 3)*x - ((27/2)*n + 21/2)*x^2)*D[P[n - 1, x], x] + ((5/2)*n^2 - n - 3/2)*x^2*P[n - 2, x])/(2*n^2 + 5*n + 3)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 5*n + 3 + (13*n^2 + 3*n - 6)*x)*v[i - 1] + ((4*n + 3)*x - ((27/2)*n + 21/2)*x^2)*deriv(v[i - 1], x) + ((5/2)*n^2 - n - 3/2)*x^2*v[i - 2])/(2*n^2 + 5*n + 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 >= 1, 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-1491609a6fc5f8c9
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-1491609a6fc5f8c9
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-1491609a6fc5f8c9

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

Used in 7 proof steps

Same fingerprint (1)

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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023