A156366 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 6\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2 - 5\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 22 + 30\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	22	30	9															61
1	160	312	180	27														679
2	1456	3696	3186	1026	81													9445
3	15904	49752	57312	29052	5832	243												158095
4	202672	753552	1087776	757296	253206	33534	729											3088765
5	2951680	12715392	22073472	19592064	9258624	2169504	195372	2187										68958295
6	48361216	236854272	480572928	519174144	318479040	108759024	18518058	1150362	6561									1731875605
7	880405504	4831109376	11219770368	14293463040	10805376672	4881245976	1251967104	158505012	6823440	19683								48328686175
8	17630351872	107141927424	280294352640	411526884096	370209407472	208760650704	72251586576	14280410160	1364779854	40665078	59049							1483501074925
9	385148108800	2567740096512	7472160981504	12425283059328	12971970415872	8786224374336	3846772316736	1047656910816	162444271392	11834521848	243045684	177147						49677478279975
10	9114999832576	66144371079168	211932071329536	393745293423360	468143895928320	370600714672128	196770921326784	68738107180608	15014499454800	1849794168240	103347205506	1455085458	531441					1802159471217925
11	232311251144704	1822809695923200	6376687738195968	13092239522032128	17469229574545920	15837644854225152	9899792864870400	4235647892488704	1203989634222624	213901576870680	21132046083744	908285813100	8719883928	1594323				70406303657894575
12	6343764407375872	53517090040375296	202966836369278976	456389628595587072	675459735899364864	690356980233985536	497014647812552448	252216377749237248	88710346261258416	20821369340590992	3040617042585792	242475846033456	8026292838726	52284228462	4782969			2947087948180076125
13	184778982658539520	1667802744510185472	6816186119416283136	16660107217670160384	27088409960484765696	30823492095842107392	25131914483531083776	14774410106000391168	6210670495418606592	1822926299975301888	357287711347558656	43234341996356352	2795982662719296	71243101284624	313590579516	14348907		131584088098220272375
14	5718564661248065536	54987967448392925184	240921738854172426240	636016182732391219200	1127073149384773386240	1413363595075289137152	1287839872865874659328	861428448063240284160	421992433921222586880	149202042862996592640	36973009064213941248	6104669168372654592	615907000022032800	32403121795851120	634602078924570	1881170405514	43046721	6242270620707298139125

Technical details

Node id: n-eab480e9d0419e9c · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-62dec3872b40fd4c

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 2)
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:           22,            30,             9;
  1:          160,           312,           180,             27;
  2:         1456,          3696,          3186,           1026,             81;
  3:        15904,         49752,         57312,          29052,           5832,           243;
  4:       202672,        753552,       1087776,         757296,         253206,         33534,           729;
  5:      2951680,      12715392,      22073472,       19592064,        9258624,       2169504,        195372,          2187;
  6:     48361216,     236854272,     480572928,      519174144,      318479040,     108759024,      18518058,       1150362,         6561;
  7:    880405504,    4831109376,   11219770368,    14293463040,    10805376672,    4881245976,    1251967104,     158505012,      6823440,       19683;
  8:  17630351872,  107141927424,  280294352640,   411526884096,   370209407472,  208760650704,   72251586576,   14280410160,   1364779854,    40665078,     59049;
  9: 385148108800, 2567740096512, 7472160981504, 12425283059328, 12971970415872, 8786224374336, 3846772316736, 1047656910816, 162444271392, 11834521848, 243045684, 177147;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+7+(3*n+6)*x)*P(n-1,x) - (2+5*x+3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 22 + 30*x + 9*x^2.

%t Mathematica

P[0, x_] := 22 + 30*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3*n + 7 + (3*n + 6)*x)*P[n - 1, x] - (2 + 5*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] = 22 + 30*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 7 + (3*n + 6)*x)*v[i - 1] - (2 + 5*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 of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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