A257617 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	44	44													92
3	8	412	1188	792												2400
4	16	3740	23540	39600	19800											86696
5	32	33724	417780	1334520	1584000	633600										4003656
6	64	303644	7055444	38214000	80883000	74131200	24710400									225297752
7	128	2733052	116227188	1005919992	3381840000	5331110400	3978374400	1136678400								14952883560
8	256	24597980	1889698580	25228249200	126603199800	305469964800	382961779200	240975820800	60243955200							1143397265816
9	512	221382844	30505755060	614264306040	4428802224000	15353691091200	28763646912000	29906008704000	16265867904000	3614637312000						98977645591656
10	1024	1992447644	490527292244	14677182630000	148220674371000	709808241542400	1864394416684800	2848334201856000	2528438799744000	1210903499520000	242180699904000					9567450235993112
11	2048	17932032892	7870353599988	346404691750392	4813549796880000	31005966516940800	109715875754284800	231027187462156800	297611163083520000	230415055453440000	98567544860928000	17921371792896000				1021432007698431720
12	4096	161388304220	126122909961620	8108974275058800	153066611200159800	1301254354626969600	6036731227349222400	16829316845267097600	29505888394338470400	32833648389484800000	22534671945367296000	8709786691347456000	1451631115224576000			119364230832779376536
13	8192	1452494754364	2019741830732340	188776620705661560	4794722692881264000	53044542679602988800	316365093833817600000	1135984795566686208000	2603295319670787456000	3904540448976820224000	3822350822161926912000	2355638872573628928000	830332997908457472000	127743538139762688000		15154281952019402897256
14	16384	13072452822044	32331846734015444	4378217629183398000	148561096304079459000	2116079205317511033600	15988801293192490963200	72488018890256606208000	211198322705940041472000	409992848363740561920000	536813469277105692672000	468949431773299276800000	262215386498591285760000	84949452862942187520000	12135636123277455360000	2077000418652515559419672

Technical details

Node id: n-a0a57351127b100c · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8c4e3f9383454e05

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:   2,         4;
  2:   4,        44,          44;
  3:   8,       412,        1188,          792;
  4:  16,      3740,       23540,        39600,         19800;
  5:  32,     33724,      417780,      1334520,       1584000,         633600;
  6:  64,    303644,     7055444,     38214000,      80883000,       74131200,       24710400;
  7: 128,   2733052,   116227188,   1005919992,    3381840000,     5331110400,     3978374400,     1136678400;
  8: 256,  24597980,  1889698580,  25228249200,  126603199800,   305469964800,   382961779200,   240975820800,    60243955200;
  9: 512, 221382844, 30505755060, 614264306040, 4428802224000, 15353691091200, 28763646912000, 29906008704000, 16265867904000, 3614637312000;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*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[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*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] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*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) = 7*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 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) = 7*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257617 → x ↦ x+1

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