row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.

palindromicγ-positivenonnegative coefficients

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

Recurrence

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

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

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	11	11	1					24
4	1	26	66	26	1				120
5	1	57	302	302	57	1			720
6	1	120	1191	2416	1191	120	1		5040
7	1	247	4293	15619	15619	4293	247	1	40320

All 15 rows b-files: A008292, A008292 (reversed triangle) ?

Technical details

Node id: n-492f384fe03c9268 · revision 0 · JSON
Definition source: derived:row_factor_parent_of:oeis_catalog:A101842
Fingerprint: f6-605cb6decd8257f5

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 ?

A008292 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A008292 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A124355, A159333, A358185, A354074, A104150, A000142, A338112, A370360

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,   11,    11,      1;
  4: 1,   26,    66,     26,       1;
  5: 1,   57,   302,    302,      57,       1;
  6: 1,  120,  1191,   2416,    1191,     120,      1;
  7: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
  8: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  9: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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

A008292 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

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

A008292 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     1;
   4: 1,   11,    11,      1;
   5: 1,   26,    66,     26,       1;
   6: 1,   57,   302,    302,      57,       1;
   7: 1,  120,  1191,   2416,    1191,     120,      1;
   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;
   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;
  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Parent-row provers of polytool, Liu–Wang theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Parent-row provers of polytool, Liu–Wang theorem
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Parent-row provers of polytool

Proof chains ? JSON

Parent-row provers of polytool polytool ? step › direct check
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Used in 2

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