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	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	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
8	1	510	18652	219210	1299268	4661550	11085604	18343050	21641590	18343050	11085604	4661550	1299268	219210	18652	510	1													92897280
9	1	1022	56993	922304	7214540	33636776	103812812	224854208	353402054	410144180	353402054	224854208	103812812	33636776	7214540	922304	56993	1022	1											1857945600
10	1	2046	173042	3821598	38886189	231006600	902181336	2477374152	4979893458	7506496404	8595133548	7506496404	4979893458	2477374152	902181336	231006600	38886189	3821598	173042	2046	1									40874803200
11	1	4094	523239	15673396	205361939	1532600742	7438347765	25319461488	63306509466	119707989084	174298076790	197346180792	174298076790	119707989084	63306509466	25319461488	7438347765	1532600742	205361939	15673396	523239	4094	1							980995276800
12	1	8190	1577928	63830130	1069144506	9919796250	59035512680	245019200790	746849420175	1728804293580	3106292464848	4392662170260	4926442358124	4392662170260	3106292464848	1728804293580	746849420175	245019200790	59035512680	9919796250	1069144506	63830130	1577928	8190	1					25505877196800
13	1	16382	4750189	258672936	5509675134	63061329372	455540215886	2276620556680	8333357924075	23192087390370	50323265843463	86585399670096	119423461968852	132847425483528	119423461968852	86585399670096	50323265843463	23192087390370	8333357924075	2276620556680	455540215886	63061329372	5509675134	258672936	4750189	16382	1			714164561510400
14	1	32766	14283358	1044618054	28184476267	395595476964	3441626698028	20511394131396	89119806963641	294120000885450	758273899616258	1557128900754738	2581171324274139	3484077275428632	3848398710032616	3484077275428632	2581171324274139	1557128900754738	758273899616258	294120000885450	89119806963641	20511394131396	3441626698028	395595476964	28184476267	1044618054	14283358	32766	1	21424936845312000

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