A257626 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 24\,x + 30\,x^{2} + 12\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	12														18
2	36	180	180													396
3	216	2052	4860	3240												10368
4	1296	21060	89100	136080	68040											315576
5	7776	205092	1385100	3645000	4082400	1632960										10958328
6	46656	1939140	19697580	79606800	150028200	132269760	44089920									427678056
7	279936	18012132	265458060	1548658440	4372250400	6378341760	4629441600	1322697600								18535139928
8	1679616	165468420	3455678700	28008121680	111222334440	238879186560	283321825920	174596083200	43649020800							883299399336
9	10077696	1509371172	43950170700	482324041800	2590172575200	7685798944320	13249461859200	13213749024000	7071141369600	1571364748800						45909482182488
10	60466176	13705272900	550042615980	8025963699600	56751911231400	223565919635520	525503656117440	754255079424000	648187958880000	306416126016000	61283225203200					2584553648562216
11	362797056	124073050212	6806090485260	130290222581640	1190079639856800	6056930181899520	18648367576977600	36129996827971200	44336056387392000	33446498678208000	14156425021939200	2573895458534400				156675470521692888
12	2176782336	1121011016580	83534181576300	2076862967459280	24157528191636840	155757445176453120	611097936758749440	1534960941664550400	2522371586944809600	2699832486326976000	1814044749239923200	694951773804288000	115825295634048000			10175161264078269096
13	13060694016	10115220537252	1019225344164300	32656559780262600	478449629766108000	3850687025304799680	18871801501974220800	59776883527705344000	126324858683274451200	179899849178803353600	170599077939389299200	103292998646444006400	36137492237822976000	5559614190434304000		704825398474324521048
14	78364164096	91193713163460	12382432438030380	508194452898896400	9297881091175458600	92347218645787385280	556891785730610890560	2180129900307270912000	5762382916912509888000	10484389935498390912000	13157660923791345561600	11193588220665666816000	6165959613078545280000	1984782265985046528000	283540323712149504000	51871491653575913390376

Technical details

Node id: n-baf71172ebf4970d · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cef044b741d89456

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:        6,         12;
  2:       36,        180,         180;
  3:      216,       2052,        4860,         3240;
  4:     1296,      21060,       89100,       136080,         68040;
  5:     7776,     205092,     1385100,      3645000,       4082400,       1632960;
  6:    46656,    1939140,    19697580,     79606800,     150028200,     132269760,       44089920;
  7:   279936,   18012132,   265458060,   1548658440,    4372250400,    6378341760,     4629441600,     1322697600;
  8:  1679616,  165468420,  3455678700,  28008121680,  111222334440,  238879186560,   283321825920,   174596083200,   43649020800;
  9: 10077696, 1509371172, 43950170700, 482324041800, 2590172575200, 7685798944320, 13249461859200, 13213749024000, 7071141369600, 1571364748800;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+24*x+30*x^2+12*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[((6 + 24*x + 30*x^2 + 12*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] = ((6 + 24*x + 30*x^2 + 12*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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.?
    
    ↑ Liu–Wang theorem (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