A257626 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-b706353c3a058005 · revision 0 · JSON
Definition source: derived:shift_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:            12,             6;
  2:           180,           180,             36;
  3:          3240,          4860,           2052,            216;
  4:         68040,        136080,          89100,          21060,          1296;
  5:       1632960,       4082400,        3645000,        1385100,        205092,          7776;
  6:      44089920,     132269760,      150028200,       79606800,      19697580,       1939140,        46656;
  7:    1322697600,    4629441600,     6378341760,     4372250400,    1548658440,     265458060,     18012132,      279936;
  8:   43649020800,  174596083200,   283321825920,   238879186560,  111222334440,   28008121680,   3455678700,   165468420,    1679616;
  9: 1571364748800, 7071141369600, 13213749024000, 13249461859200, 7685798944320, 2590172575200, 482324041800, 43950170700, 1509371172, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+9+(3*n+3)*x)*P(n-1,x) - (3*x+3*x^2)*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[(3*n + 9 + (3*n + 3)*x)*P[n - 1, x] - (3*x + 3*x^2)*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] = (3*n + 9 + (3*n + 3)*x)*v[i - 1] - (3*x + 3*x^2)*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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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