A257625 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	72	72	9													153
3	1296	1944	702	27												3969
4	31104	62208	37584	6480	81											137457
5	933120	2332800	1982880	641520	58806	243										5949369
6	33592320	100776960	110574720	53187840	10328472	530712	729									308991753
7	1410877440	4938071040	6642881280	4262025600	1302857136	161295624	4780782	2187								18722791089
8	67722117120	270888468480	431728496640	347075850240	146343262464	30263321088	2476803744	43040160	6561							1296541366497
9	3656994324480	16456474460160	30322577940480	29332141977600	15843800931840	4677587792640	680112210240	37668538080	387400806	19683						101007745596009
10	219419659468800	1097098297344000	2294763938611200	2596465970380800	1714330370672640	663173064529920	142617563447040	14960390100480	569676880872	3486725352	59049					8743402418220153
11	14481697524940800	79649336387174400	186451855633612800	241663327447449600	188842844235386880	90733114779310080	26163238032898560	4209723575481600	324422375965776	8586993917304	31380882462	177147				832528178367197409
12	1042682221795737600	6256093330774425600	16205019530408755200	23677575453278208000	21389286223118499840	12308718811323432960	4480468045577441280	989678562350100480	121448673561181824	6967435785792768	129181479349104	282429005040	531441			86478067751882461137
13	81329213300067532800	528639886450438963200	1501201728830413209600	2441570757612444057600	2504643256885256847360	1681672100322568273920	742072004423807016960	210366682022123827200	36302852764388770560	3446332645010935680	148641418129932000	1941111338297040	2541864234006	1594323		9731395400327853492249
14	6831653917205672755200	47821577420439709286400	147734515959572673331200	264726589291719819264000	304097553142699386470400	233574959059878327091200	121384733328793718784000	42228640844000516505600	9511203298380825000960	1301384120575157498880	96618375450413631360	3156409784817918720	29147172445263672	22876787671992	4782969	1179312614338550275256553

Technical details

Node id: n-7bc03246f43906fd · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-2a2e8db5a042c827

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,              3;
  2:            72,             72,              9;
  3:          1296,           1944,            702,             27;
  4:         31104,          62208,          37584,           6480,             81;
  5:        933120,        2332800,        1982880,         641520,          58806,           243;
  6:      33592320,      100776960,      110574720,       53187840,       10328472,        530712,          729;
  7:    1410877440,     4938071040,     6642881280,     4262025600,     1302857136,     161295624,      4780782,        2187;
  8:   67722117120,   270888468480,   431728496640,   347075850240,   146343262464,   30263321088,   2476803744,    43040160,      6561;
  9: 3656994324480, 16456474460160, 30322577940480, 29332141977600, 15843800931840, 4677587792640, 680112210240, 37668538080, 387400806, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (6*n+(6*n-3)*x)*P(n-1,x) - (6*x+6*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[(6*n + (6*n - 3)*x)*P[n - 1, x] - (6*x + 6*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] = (6*n + (6*n - 3)*x)*v[i - 1] - (6*x + 6*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), 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 ?

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