A257620 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	54	54													117
3	27	378	972	648												2025
4	81	2430	12150	19440	9720											43821
5	243	15066	131220	379080	437400	174960										1137969
6	729	91854	1316574	6123600	12247200	11022480	3674160									34476597
7	2187	555498	12675852	89282088	275562000	418854240	308629440	88179840								1193741145
8	6561	3346110	119082150	1223495280	5472661320	12499492320	15277157280	9523422720	2380855680							46499519421
9	19683	20116026	1101854340	16110929160	100442349000	323498765520	583309641600	595213920000	321415516800	71425670400						2012538782529
10	59049	120814254	10097733294	206553402000	1748299172400	7630940061360	19042976549520	28284565478400	24748994793600	11785235616000	2357047123200					95814830803077
11	177147	725239818	91966927932	2599813623528	29322788616000	168826306207680	560152248828480	1135861008670080	1431906127344000	1096026912288000	466695330393600	84853696435200				4976336924751465
12	531441	4352501790	834229509750	32301366617520	478839033592920	3566683706826240	15308549655759360	40704318179965440	69329712672380160	75837991188960000	51569834008492800	19855764965836800	3309294160972800			279994127521947021
13	1594323	26118199386	7547238103860	397627153527240	7667106003156600	72819409327544880	396379900614297600	1344308828057395200	2970918833013331200	4355031115840204800	4204458231515942400	2571321563075865600	903437305945574400	138990354760857600		16965737848665595089
14	4782969	156718762254	68160206729214	4862092699573200	120970997350257600	1448757275952626640	9853185508778692080	41776529488120627200	116511146848909612800	219778498465606080000	282463148462752857600	243928072605305088000	135515595891836160000	43781961749670144000	6254565964238592000	1101437363668150585557

Technical details

Node id: n-e62f038d3232303b · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-a83eeccfb5efd735

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,       54,         54;
  3:    27,      378,        972,         648;
  4:    81,     2430,      12150,       19440,         9720;
  5:   243,    15066,     131220,      379080,       437400,       174960;
  6:   729,    91854,    1316574,     6123600,     12247200,     11022480,      3674160;
  7:  2187,   555498,   12675852,    89282088,    275562000,    418854240,    308629440,     88179840;
  8:  6561,  3346110,  119082150,  1223495280,   5472661320,  12499492320,  15277157280,   9523422720,   2380855680;
  9: 19683, 20116026, 1101854340, 16110929160, 100442349000, 323498765520, 583309641600, 595213920000, 321415516800, 71425670400;

%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) + (3*x+9*x^2+9*x^3+3*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] + (3*x + 9*x^2 + 9*x^3 + 3*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] + (3*x + 9*x^2 + 9*x^3 + 3*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
  - Parent-row provers of polytool polytool ? 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)
    
    ↑ Parent-row provers of polytool (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

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