A069466: Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.

A069466 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A069466

\(\left(n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x - 8\,n - 8\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-12 + 24\,x - 12\,x^{2} + 32\,n - 64\,n x + 32\,n x^{2} - 16\,n^{2} + 32\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	6	24	6						36
3	20	180	180	20					400
4	70	1120	2520	1120	70				4900
5	252	6300	25200	25200	6300	252			63504
6	924	33264	207900	369600	207900	33264	924		853776
7	3432	168168	1513512	4204200	4204200	1513512	168168	3432	11778624

Show all 15 stored rows

Generated b-files: A002894 (row sums), A069466, A069466 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-fdc03fad975441d0 · revision 1 · JSON
Definition source: oeis_catalog:A069466
Fingerprint: f6-b8a804edad022635

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: 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): no

OEIS matches ?

A069466: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A069466: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A002894: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); 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.

A069466: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); 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.

A069466: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); 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.

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-412107fab2ca179a
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-7dc194e46d1f26fe
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-7dc194e46d1f26fe
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-7dc194e46d1f26fe
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-7dc194e46d1f26fe

Proof chains

2 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

theorem_check three_term_recurrence e-412107fab2ca179a root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-7dc194e46d1f26fe root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 6 proof steps

e-1435140a6a62e1e4 basis_map (A013610) → basis_map_A013610 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-2cd9cd27e716475b basis_map (A007318) → basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-621bc4ded5b72353 basis_map (A013611) → basis_map_A013611 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-62dc2e1e9139053e basis_map (A008277) → basis_map_A008277 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-aca51a80a394c762 basis_map (A013609) → basis_map_A013609 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-510754e51c3ada00 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and en…

Related sequences ?

Same fingerprint (1)

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

A008459: Square the entries of Pascal's triangle.

References

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