A102413: Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).

A102413 ↗ palindromicγ-positivenonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing interlacing (weak)

Recurrence from A102413

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 1 + 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	6	6	1					14
4	1	8	16	8	1				34
5	1	10	30	30	10	1			82
6	1	12	48	76	48	12	1		198
7	1	14	70	154	154	70	14	1	478

All 15 rows b-files: A099425 (row sums), A102413, A102413 (reversed triangle) ?

Other recurrences

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

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 + n - 2\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 9\,x^{2} + 2\,x^{3} - n x - 6\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 6\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

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

Technical details

Node id: n-9d2e609ab19d361f · revision 1 · JSON
Definition source: oeis_catalog:A102413
derived:reversal (equal by e-426fd541940deab2)
Fingerprint: f6-2f47db79ca6a11a0

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

Matches ?

A102413 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
A102413 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A099425, A002203

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   1;
  3: 1,  6,   6,   1;
  4: 1,  8,  16,   8,   1;
  5: 1, 10,  30,  30,  10,   1;
  6: 1, 12,  48,  76,  48,  12,   1;
  7: 1, 14,  70, 154, 154,  70,  14,   1;
  8: 1, 16,  96, 272, 384, 272,  96,  16,  1;
  9: 1, 18, 126, 438, 810, 810, 438, 126, 18, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*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] = 1 + x; v[3] = 1 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (n - 2)*x*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)

polytool c48629374597

A102413 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   1;
  3: 1,  6,   6,   1;
  4: 1,  8,  16,   8,   1;
  5: 1, 10,  30,  30,  10,   1;
  6: 1, 12,  48,  76,  48,  12,   1;
  7: 1, 14,  70, 154, 154,  70,  14,   1;
  8: 1, 16,  96, 272, 384, 272,  96,  16,  1;
  9: 1, 18, 126, 438, 810, 810, 438, 126, 18, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*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] = 1 + x; v[3] = 1 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (n - 2)*x*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)

polytool c48629374597

A102413 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   1;
  3: 1,  6,   6,   1;
  4: 1,  8,  16,   8,   1;
  5: 1, 10,  30,  30,  10,   1;
  6: 1, 12,  48,  76,  48,  12,   1;
  7: 1, 14,  70, 154, 154,  70,  14,   1;
  8: 1, 16,  96, 272, 384, 272,  96,  16,  1;
  9: 1, 18, 126, 438, 810, 810, 438, 126, 18, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*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] = 1 + x; v[3] = 1 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (n - 2)*x*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row, Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row, Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
↑ Liu–Wang theorem for three-term recurrences (above)
↑ Liu–Wang theorem with a lag row (above)

Used in 11

3 more

References

[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

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	6	6	1					14
4	1	8	16	8	1				34
5	1	10	30	30	10	1			82
6	1	12	48	76	48	12	1		198
7	1	14	70	154	154	70	14	1	478

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	6	6	1					14
4	1	8	16	8	1				34
5	1	10	30	30	10	1			82
6	1	12	48	76	48	12	1		198
7	1	14	70	154	154	70	14	1	478

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	4	1						6
3	1	6	6	1					14
4	1	8	16	8	1				34
5	1	10	30	30	10	1			82
6	1	12	48	76	48	12	1		198
7	1	14	70	154	154	70	14	1	478