A165891: Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)(1-x)^n(1+x)^(n+2)( Sum_{j >= 0} j^(n+1)x^j ), read by rows.

A165891 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A165891

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2	1													4
2	1	6	10	6	1											24
3	1	14	47	68	47	14	1									192
4	1	30	176	450	606	450	176	30	1							1920
5	1	62	597	2392	5162	6612	5162	2392	597	62	1					23040
6	1	126	1926	11382	35967	69132	85492	69132	35967	11382	1926	126	1			322560
7	1	254	6043	50892	223785	600546	1060411	1277096	1060411	600546	223785	50892	6043	254	1	5160960

All 15 rows b-files: A165891, A165891 (reversed triangle) ?

Technical details

Node id: n-b3367e055aead099 · revision 0 · JSON
Definition source: oeis_catalog:A165891
Fingerprint: f6-394a08467f0bbaf0

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A165891 triangle 64 terms, 8 rows · tabf · nonn · catalog b-file?
A165891 reversed triangle 64 terms, 8 rows · tabf · nonn · catalog b-file?

Row sums ? A052594, A121892, A032107, A002866, A052656

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,    2,     1;
  2: 1,    6,    10,      6,       1;
  3: 1,   14,    47,     68,      47,       14,         1;
  4: 1,   30,   176,    450,     606,      450,       176,        30,         1;
  5: 1,   62,   597,   2392,    5162,     6612,      5162,      2392,       597,        62,         1;
  6: 1,  126,  1926,  11382,   35967,    69132,     85492,     69132,     35967,     11382,      1926,       126,         1;
  7: 1,  254,  6043,  50892,  223785,   600546,   1060411,   1277096,   1060411,    600546,    223785,     50892,      6043,      254,       1;
  8: 1,  510, 18652, 219210, 1299268,  4661550,  11085604,  18343050,  21641590,  18343050,  11085604,   4661550,   1299268,   219210,   18652,    510,     1;
  9: 1, 1022, 56993, 922304, 7214540, 33636776, 103812812, 224854208, 353402054, 410144180, 353402054, 224854208, 103812812, 33636776, 7214540, 922304, 56993, 1022, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x+(2*n-1)*x^2)*P(n-1,x) + (x-x^3)*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 + 2*x + (2*n - 1)*x^2)*P[n - 1, x] + (x - x^3)*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 + 2*x + (2*n - 1)*x^2)*v[i - 1] + (x - x^3)*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

A165891 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    2,     1;
  2: 1,    6,    10,      6,       1;
  3: 1,   14,    47,     68,      47,       14,         1;
  4: 1,   30,   176,    450,     606,      450,       176,        30,         1;
  5: 1,   62,   597,   2392,    5162,     6612,      5162,      2392,       597,        62,         1;
  6: 1,  126,  1926,  11382,   35967,    69132,     85492,     69132,     35967,     11382,      1926,       126,         1;
  7: 1,  254,  6043,  50892,  223785,   600546,   1060411,   1277096,   1060411,    600546,    223785,     50892,      6043,      254,       1;
  8: 1,  510, 18652, 219210, 1299268,  4661550,  11085604,  18343050,  21641590,  18343050,  11085604,   4661550,   1299268,   219210,   18652,    510,     1;
  9: 1, 1022, 56993, 922304, 7214540, 33636776, 103812812, 224854208, 353402054, 410144180, 353402054, 224854208, 103812812, 33636776, 7214540, 922304, 56993, 1022, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x+(2*n-1)*x^2)*P(n-1,x) + (x-x^3)*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 + 2*x + (2*n - 1)*x^2)*P[n - 1, x] + (x - x^3)*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 + 2*x + (2*n - 1)*x^2)*v[i - 1] + (x - x^3)*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

A165891 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,    2,     1;
  2: 1,    6,    10,      6,       1;
  3: 1,   14,    47,     68,      47,       14,         1;
  4: 1,   30,   176,    450,     606,      450,       176,        30,         1;
  5: 1,   62,   597,   2392,    5162,     6612,      5162,      2392,       597,        62,         1;
  6: 1,  126,  1926,  11382,   35967,    69132,     85492,     69132,     35967,     11382,      1926,       126,         1;
  7: 1,  254,  6043,  50892,  223785,   600546,   1060411,   1277096,   1060411,    600546,    223785,     50892,      6043,      254,       1;
  8: 1,  510, 18652, 219210, 1299268,  4661550,  11085604,  18343050,  21641590,  18343050,  11085604,   4661550,   1299268,   219210,   18652,    510,     1;
  9: 1, 1022, 56993, 922304, 7214540, 33636776, 103812812, 224854208, 353402054, 410144180, 353402054, 224854208, 103812812, 33636776, 7214540, 922304, 56993, 1022, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x+(2*n-1)*x^2)*P(n-1,x) + (x-x^3)*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 + 2*x + (2*n - 1)*x^2)*P[n - 1, x] + (x - x^3)*D[P[n - 1, x], x]];
Table[Reverse[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 + 2*x + (2*n - 1)*x^2)*v[i - 1] + (x - x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); 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 Product of real-rooted factors (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Product of real-rooted factors (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Product of real-rooted factors (elementary)

Proof chains ? JSON

Product of real-rooted factors (elementary) polytool ? step ›
- from 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 o…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

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