A249247: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

A249247 ↗ nonnegative coefficients

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

Recurrence from A249247

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	3							5
3	6	10	1						17
4	24	42	7						73
5	120	216	45	1					382
6	720	1320	312	13					2365
7	5040	9360	2400	136	1				16937
8	40320	75600	20520	1400	21				137861
9	362880	685440	194040	15000	325	1			1257686
10	3628800	6894720	2016000	170520	4650	31			12714721
11	39916800	76204800	22861440	2069760	66150	666	1		141119617
12	479001600	918086400	281232000	26853120	964320	12642	43		1706150125
13	6227020800	11975040000	3732220800	371952000	14605920	230496	1225	1	22321071242
14	87178291200	168129561600	53169177600	5488560000	231336000	4191264	29792	57	314201147513

b-files: A249247 ?

Technical details

Node id: n-bf8bba6616804a77 · revision 0 · JSON
Definition source: oeis_catalog:A249247
Fingerprint: f6-c65a80d0536e3dd0

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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

Matches ?

A249247 triangle 46 terms, 11 rows · tabf · nonn · catalog b-file?

Row sums ? A102038

Submission text ?

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

%e Example

Triangle begins:
  0:      1;
  1:      1,      1;
  2:      2,      3;
  3:      6,     10,      1;
  4:     24,     42,      7;
  5:    120,    216,     45,     1;
  6:    720,   1320,    312,    13;
  7:   5040,   9360,   2400,   136,   1;
  8:  40320,  75600,  20520,  1400,  21;
  9: 362880, 685440, 194040, 15000, 325, 1;

%F Formula

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

A249247 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:      1;
   2:      1,      1;
   3:      2,      3;
   4:      6,     10,      1;
   5:     24,     42,      7;
   6:    120,    216,     45,     1;
   7:    720,   1320,    312,    13;
   8:   5040,   9360,   2400,   136,   1;
   9:  40320,  75600,  20520,  1400,  21;
  10: 362880, 685440, 194040, 15000, 325, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n-1)*P(n-1,x) + x*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[(n - 1)*P[n - 1, x] + x*P[n - 2, x]];
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] = (n - 1)*v[i - 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, 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 10

2 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