A257621 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	6														9
2	9	60	60													129
3	27	474	1260	840												2601
4	81	3480	18600	30240	15120											67521
5	243	24846	239400	714000	831600	332640										2142729
6	729	175380	2881860	14061600	28652400	25945920	8648640									80366529
7	2187	1232034	33454260	251270040	797504400	1227108960	908107200	259459200								3478138281
8	6561	8637360	380317200	4237410240	19675444320	45768602880	56423727360	35286451200	8821612800							170602209921
9	19683	60500886	4269862800	68885594400	450106826400	1485537641280	2713424313600	2786591808000	1508495788800	335221286400						9352593642249
10	59049	423624300	47573499660	1093061995200	9791970400800	44069715930240	111886435140480	167787075456000	147541474080000	70396470144000	14079294028800					566693494358529
11	177147	2965724394	527544739260	17061958923240	205722553528800	1229026815213120	4166746362979200	8557992393350400	10868712158304000	8352038350656000	3562061389286400	647647525324800				37607540018206761
12	531441	20761133640	5832649375800	263315010198240	4215843777665520	32793512927535360	144456848995979520	390300155083238400	671376666914553600	738740557691136000	503954250466867200	194294257597440000	32382376266240000			2712783638141894721
13	1594323	145331124126	64366754470200	4031382244234800	84840701959213200	846999360441954720	4752966259007366400	16433010277459776000	36768388614839481600	54323195092707340800	52697136193103001600	32313725628555571200	11366214069450240000	1748648318376960000		211339220410232328969
14	4782969	1017327434820	709487610413460	61371868226104800	1684538217621277200	21347480733267648960	150352072364689715520	652012306371474048000	1845615950953014240000	3515803375979486592000	4547551050197137382400	3942813369424849920000	2195363198969740800000	709951217261045760000	101421602465863680000	17683978245311359800129

Technical details

Node id: n-3447ba8ca39d0e59 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b4bba0102ede147f

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:     1;
  1:     3,        6;
  2:     9,       60,         60;
  3:    27,      474,       1260,         840;
  4:    81,     3480,      18600,       30240,        15120;
  5:   243,    24846,     239400,      714000,       831600,        332640;
  6:   729,   175380,    2881860,    14061600,     28652400,      25945920,       8648640;
  7:  2187,  1232034,   33454260,   251270040,    797504400,    1227108960,     908107200,     259459200;
  8:  6561,  8637360,  380317200,  4237410240,  19675444320,   45768602880,   56423727360,   35286451200,    8821612800;
  9: 19683, 60500886, 4269862800, 68885594400, 450106826400, 1485537641280, 2713424313600, 2786591808000, 1508495788800, 335221286400;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*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[Together[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257621 → x ↦ x+1

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