A256890 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 20 + 20\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	20	20	4															44
2	120	180	76	8														384
3	840	1680	1100	260	16													3896
4	6720	16800	15000	5700	844	32												45096
5	60480	181440	205800	109200	27020	2660	64											586664
6	604800	2116800	2916480	1999200	708120	121380	8236	128										8475144
7	6652800	26611200	43182720	36408960	16952040	4268880	526700	25220	256									134628776
8	79833600	359251200	671328000	673142400	390479040	131594400	24504600	2232900	76684	512								2332443336
9	1037836800	5189184000	10977120000	12773376000	8899450560	3786108480	961098600	135920400	9315020	232100	1024							43769642984
10	14529715200	79913433600	188806464000	250278336000	203954889600	105270580800	34191548160	6718034400	735492120	38420580	700396	2048						884437616904
11	217945728000	1307674368000	3413445235200	5080211136000	4746287145600	2888299814400	1149888579840	293085112320	45456651240	3907984080	157184300	2109380	4096					19146361052456
12	3487131648000	22666355712000	64788000076800	107004087590400	112837768243200	79234169414400	37493584128000	11836874649600	2415248996160	300095796000	20483026200	639284100	6344524	8192				442084444917576
13	59281238016000	414968666112000	1289148981120000	2340301227264000	2750940038246400	2192025483648000	1204771119552000	455811003648000	116432238162240	19307509341120	1943955959400	106250835600	2588859020	19066340	16384			10845040319846504
14	1067062284288000	8002967132160000	26850913689600000	53152604144640000	68937076489881600	61507611746380800	38569533754752000	17061632235648000	5266621414454400	1105225489368000	150704213063040	12407491605600	546787332120	10450767780	57264556	32768		281684917381238664
15	20274183401472000	162193467211776000	584513006837760000	1253205371658240000	1778192605373184000	1757288558817792000	1239495279768345600	629004458130048000	228549157915478400	58451847623769600	10198141832511360	1154189424286080	78272460958440	2796641267280	42089393900	171924740	65536	7722601379358272936

Technical details

Node id: n-a62fc9bae1fba01f · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-cef044b741d89456

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          20,          20,            4;
   2:         120,         180,           76,            8;
   3:         840,        1680,         1100,          260,           16;
   4:        6720,       16800,        15000,         5700,          844,           32;
   5:       60480,      181440,       205800,       109200,        27020,         2660,          64;
   6:      604800,     2116800,      2916480,      1999200,       708120,       121380,        8236,        128;
   7:     6652800,    26611200,     43182720,     36408960,     16952040,      4268880,      526700,      25220,       256;
   8:    79833600,   359251200,    671328000,    673142400,    390479040,    131594400,    24504600,    2232900,     76684,      512;
   9:  1037836800,  5189184000,  10977120000,  12773376000,   8899450560,   3786108480,   961098600,  135920400,   9315020,   232100,   1024;
  10: 14529715200, 79913433600, 188806464000, 250278336000, 203954889600, 105270580800, 34191548160, 6718034400, 735492120, 38420580, 700396, 2048;

%F Formula

The row polynomials satisfy P(n,x) = (n+4+(n+2)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 20 + 20*x + 4*x^2.

%t Mathematica

P[1, x_] := 20 + 20*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + (n + 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 20 + 20*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = (n + 4 + (n + 2)*x)*v[i - 1] - (x + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle 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) = x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle 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) = x + 2.?
    
    ↑ 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