Real-rooted atlas

A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.

A104684 nonnegative coefficients

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

Recurrence from OEIS A104684
\(\left(1 - n\right)\,P_{n}(x) = \left(6 + 3\,x - 4\,n - 2\,n x\right)\,P_{n-1}(x) + \left(-2\,x^{2} + n x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
366113
4203012163
57014090201321
62526305602103011683
79242772315016804204218989
83432120121663211550420075656148639

Show all 15 stored rows

Generated b-files: A001850 (row sums), A063007 (reversed triangle), A104684 ?

Computed in your browser ?

Other recurrences

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

\(\left(-1 + n\right)\,P_{n}(x) = \left(-6 - 5\,x + 4\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-78ba05b4a3dd925d · revision 2 · JSON
Definition source
oeis_catalog:A104684
derived:reversal (equal by e-0431973470657681)
Fingerprint
f6-fd15deff59829444

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:     2,      1;
   3:     6,      6,      1;
   4:    20,     30,     12,      1;
   5:    70,    140,     90,     20,      1;
   6:   252,    630,    560,    210,     30,     1;
   7:   924,   2772,   3150,   1680,    420,    42,     1;
   8:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;
   9: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;
  10: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = -(4*n-6+(2*n-3)*x)*P(n-1,x) + (n-2)*x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 6 + (2*n - 3)*x)*P[n - 1, x] + (n - 2)*x^2*P[n - 2, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 6 + (2*n - 3)*x)*v[i - 1] + (n - 2)*x^2*v[i - 2])/(-n + 1)); 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.

A063007: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  6,    6;
  3: 1, 12,   30,    20;
  4: 1, 20,   90,   140,    70;
  5: 1, 30,  210,   560,   630,    252;
  6: 1, 42,  420,  1680,  3150,   2772,    924;
  7: 1, 56,  756,  4200, 11550,  16632,  12012,   3432;
  8: 1, 72, 1260,  9240, 34650,  72072,  84084,  51480,  12870;
  9: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where -n*P(n,x) = -(4*n-2+(2*n-1)*x)*P(n-1,x) + (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (n - 1)*x^2*P[n - 2, x])/(-n)]];
Table[Reverse[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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (2*n - 1)*x)*v[i - 1] + (n - 1)*x^2*v[i - 2])/(-n)); v};
row(n, p) = {my(c = Vecrev(p)); 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.

A104684: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:     1;
  1:     2,      1;
  2:     6,      6,      1;
  3:    20,     30,     12,      1;
  4:    70,    140,     90,     20,      1;
  5:   252,    630,    560,    210,     30,     1;
  6:   924,   2772,   3150,   1680,    420,    42,     1;
  7:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;
  8: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;
  9: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;
%F Formula
The row polynomials satisfy -n*P(n,x) = -(4*n-2+(2*n-1)*x)*P(n-1,x) + (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (n - 1)*x^2*P[n - 2, x])/(-n)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (2*n - 1)*x)*v[i - 1] + (n - 1)*x^2*v[i - 2])/(-n)); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-0e64edb308434359
  • 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-74c20febd3423e2e
  • 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-5047c71cb920acb6, e-74c20febd3423e2e
  • 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-5047c71cb920acb6, e-74c20febd3423e2e
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-74c20febd3423e2e
  • 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-5047c71cb920acb6

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 12 proof steps

Same fingerprint (4)

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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023