A131222: Exponential Riordan array [1, log((1-x)/(1-2x))].

A131222 ↗ nonnegative coefficients

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

Recurrence from A131222

\(P_{n}(x) = \left(-3 + x + 3\,n\right)\,P_{n-1}(x) + \left(-4 + 6\,n - 2\,n^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	3	1													4
3	0	14	9	1												24
4	0	90	83	18	1											192
5	0	744	870	275	30	1										1920
6	0	7560	10474	4275	685	45	1									23040
7	0	91440	143892	70924	14805	1435	63	1								322560
8	0	1285200	2233356	1274196	324289	41160	2674	84	1							5160960
9	0	20603520	38769840	24870572	7398972	1151409	98280	4578	108	1						92897280
10	0	371226240	745785936	526791060	177945200	32559975	3419913	209790	7350	135	1					1857945600
11	0	7428153600	15766233120	12072814776	4533332100	947490830	117466965	8889573	410850	11220	165	1				40874803200
12	0	163459296000	363640940640	298275087528	122524830076	28637334990	4071519815	364669074	20830623	751410	16445	198	1			980995276800
13	0	3923502105600	9092247615360	7914320990784	3512369295864	903331310596	144200769570	14852759207	1006107102	44919303	1299870	23309	234	1		25505877196800
14	0	102017281766400	245065185624960	224688938947200	106668976545768	29807441892228	5256847141546	609681627555	47591781809	2523519999	90483393	2147145	32123	273	1	714164561510400

b-files: A002866 (row sums), A079638, A131222 ?

Technical details

Node id: n-e41c30a965e7cb9c · revision 0 · JSON
Definition source: oeis_catalog:A131222
Fingerprint: f6-bd19e0a95c26bd62

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 ?

A131222 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A079638 triangle 43 terms, 8 rows · tabl · nonn · catalog b-file?

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

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,        1;
  2: 0,        3,        1;
  3: 0,       14,        9,        1;
  4: 0,       90,       83,       18,       1;
  5: 0,      744,      870,      275,      30,       1;
  6: 0,     7560,    10474,     4275,     685,      45,     1;
  7: 0,    91440,   143892,    70924,   14805,    1435,    63,    1;
  8: 0,  1285200,  2233356,  1274196,  324289,   41160,  2674,   84,   1;
  9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*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

A079638 triangle, offset 1, from P₁, low zeros dropped

%e Example

Triangle begins:
   1:         1;
   2:         3,         1;
   3:        14,         9,         1;
   4:        90,        83,        18,         1;
   5:       744,       870,       275,        30,        1;
   6:      7560,     10474,      4275,       685,       45,       1;
   7:     91440,    143892,     70924,     14805,     1435,      63,      1;
   8:   1285200,   2233356,   1274196,    324289,    41160,    2674,     84,    1;
   9:  20603520,  38769840,  24870572,   7398972,  1151409,   98280,   4578,  108,   1;
  10: 371226240, 745785936, 526791060, 177945200, 32559975, 3419913, 209790, 7350, 135, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 3*x + x^2.

%t Mathematica

P[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = x; v[2] = 3*x + x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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

A131222 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,        1;
  2: 0,        3,        1;
  3: 0,       14,        9,        1;
  4: 0,       90,       83,       18,       1;
  5: 0,      744,      870,      275,      30,       1;
  6: 0,     7560,    10474,     4275,     685,      45,     1;
  7: 0,    91440,   143892,    70924,   14805,    1435,    63,    1;
  8: 0,  1285200,  2233356,  1274196,  324289,   41160,  2674,   84,   1;
  9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 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