A079510: Triangle T(n,k) read by rows; related to number of preorders.

A079510 ↗ nonnegative coefficients

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

Recurrence from A079510

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	3	6						9
3	0	0	20	24					44
4	0	0	15	130	120				265
5	0	0	0	210	924	720			1854
6	0	0	0	105	2380	7308	5040		14833
7	0	0	0	0	2520	26432	64224	40320	133496

All 15 rows b-files: A008306 (reversed triangle), A079510 ?

Technical details

Node id: n-f364085508d20047 · revision 0 · JSON
Definition source: oeis_catalog:A079510
Fingerprint: f6-adb63b75c11d63d7

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

Matches ?

A079510 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A008306 reversed triangle 36 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A000166, A257953, A260216, A260115

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 3,  6;
  3: 0, 0, 20,  24;
  4: 0, 0, 15, 130,  120;
  5: 0, 0,  0, 210,  924,   720;
  6: 0, 0,  0, 105, 2380,  7308,   5040;
  7: 0, 0,  0,   0, 2520, 26432,  64224,   40320;
  8: 0, 0,  0,   0,  945, 44100, 303660,  623376,  362880;
  9: 0, 0,  0,   0,    0, 34650, 705320, 3678840, 6636960, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (n+1)*x*P(n-1,x) + (n+1)*x*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[(n + 1)*x*P[n - 1, x] + (n + 1)*x*P[n - 2, 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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1)*x*v[i - 1] + (n + 1)*x*v[i - 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

A008306 reversed triangle, offset 2, from P₀, low zeros dropped

%e Example

Triangle begins:
   2:       1;
   3:       2;
   4:       6,       3;
   5:      24,      20;
   6:     120,     130,      15;
   7:     720,     924,     210;
   8:    5040,    7308,    2380,    105;
   9:   40320,   64224,   26432,   2520;
  10:  362880,  623376,  303660,  44100,   945;
  11: 3628800, 6636960, 3678840, 705320, 34650;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (n-1)*x*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 2*x.

%t Mathematica

P[2, x_] := 1;
P[3, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[(n - 1)*x*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 2*x; for(n = 4, N, my(i = n - 1); v[i] = (n - 1)*x*v[i - 1] + (n - 1)*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)

polytool c48629374597

A079510 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 0, 2;
   3: 0, 3,  6;
   4: 0, 0, 20,  24;
   5: 0, 0, 15, 130,  120;
   6: 0, 0,  0, 210,  924,   720;
   7: 0, 0,  0, 105, 2380,  7308,   5040;
   8: 0, 0,  0,   0, 2520, 26432,  64224,   40320;
   9: 0, 0,  0,   0,  945, 44100, 303660,  623376,  362880;
  10: 0, 0,  0,   0,    0, 34650, 705320, 3678840, 6636960, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + n*x*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[n*x*P[n - 1, x] + n*x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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] = n*x*v[i - 1] + n*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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
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
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 12

4 more

Related ?

Same fingerprint ?

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