A257612 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	32	32													68
3	8	208	576	384												1176
4	16	1280	7424	12288	6144											27152
5	32	7744	84480	261120	307200	122880										783456
6	64	46592	906752	4669440	9707520	8847360	2949120									27126848
7	128	279808	9440256	76253184	249446400	388792320	289013760	82575360								1095801216
8	256	1679360	96641024	1180827648	5710086144	13542359040	16845373440	10569646080	2642411520							50589024512
9	512	10077184	979845120	17691279360	121674792960	412133621760	762996326400	788759838720	428070666240	95126814720						2627443262976
10	1024	60465152	9879068672	259436052480	2473206743040	11500435537920	29729111408640	45026692300800	39794717491200	19025362944000	3805072588800					151623974601728
11	2048	362795008	99274407936	3750653558784	48668698214400	302473716695040	1048967349534720	2183215888465920	2793874548326400	2155573621555200	920827566489600	167423193907200				9624874873952256
12	4096	2176778240	995646439424	53700442718208	936047024799744	7627795731578880	34532520288583680	94867562440949760	164854643074007040	182491281358848000	124897702654771200	48217879845273600	8036313307545600			666516443992297472
13	8192	13060685824	9973878620160	763753955328000	17708053529886720	186532446590730240	1080912625061068800	3812937441308835840	8640819862626631680	12869435842300477440	12545354765854310400	7713521389692518400	2716273897950412800	417888291992371200		50002158357801885696
14	16384	78364147712	99843271688192	10812241918033920	330965026823208960	4457874895593799680	32580506969765314560	144653676740975001600	415801873451188224000	800108077061976883200	1041682333857900134400	906817593623445504000	506062721602761523200	163812210461009510400	23401744351572787200	4039720490206565777408

Technical details

Node id: n-9b0cf606b905d184 · revision 0 · JSON
Definition source: derived:mobius_x_over_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:   2,        4;
  2:   4,       32,        32;
  3:   8,      208,       576,         384;
  4:  16,     1280,      7424,       12288,         6144;
  5:  32,     7744,     84480,      261120,       307200,       122880;
  6:  64,    46592,    906752,     4669440,      9707520,      8847360,      2949120;
  7: 128,   279808,   9440256,    76253184,    249446400,    388792320,    289013760,     82575360;
  8: 256,  1679360,  96641024,  1180827648,   5710086144,  13542359040,  16845373440,  10569646080,   2642411520;
  9: 512, 10077184, 979845120, 17691279360, 121674792960, 412133621760, 762996326400, 788759838720, 428070666240, 95126814720;

%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) + (4*x+12*x^2+12*x^3+4*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] + (4*x + 12*x^2 + 12*x^3 + 4*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] + (4*x + 12*x^2 + 12*x^3 + 4*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) = 4*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) = 4*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