A257614 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	36	36	4													76
3	504	756	268	8												1536
4	9576	19152	11484	1908	16											42136
5	229824	574560	486360	154980	13420	32										1459176
6	6664896	19994688	21881160	10437840	1980540	94068	64									61053256
7	226606464	793122624	1065004416	679704480	205170840	24613092	658732	128								2994880776
8	8837652096	35350608384	56265052032	45068026752	18851744520	3832487568	301285692	4611636	256							168511468936
9	388856692224	1749855115008	3220984279296	3107454440832	1669278590784	487555643232	69370288344	3656933028	32282476	512						10697044265736
10	19053977918976	95269889594880	199112301722880	224829869322240	147847621239360	56771857388736	12044259629640	1230491964240	44173738620	225979380	1024					756204668499976
11	1028914807624704	5659031441935872	13238626086766080	17131081575928320	13345524287061120	6377507743932864	1821902380607232	288353059172640	21536795732760	532118677860	1581859756	2048				58913011879301256
12	60705973649857536	364235841899145216	942962313232203264	1375983015418851840	1239980873384373120	710619661914978816	256915416843461376	56111837696538624	6752966420720520	373575188946960	6399660872124	11073026484	4096			5014647886382979976
13	3885182313590882304	25253685038340734976	71681407902790253568	116457208043598309888	119230360153598453760	79793363496827844864	35028051663531902976	9848536632190384128	1677090811903835328	155663189845843680	6440373464308056	76895587703844	77511201772	8192		463017066592781667336
14	268077579637770878976	1876543057464396152832	5794931538641750529024	10374529484813353187328	11898062465793436348416	9114734921079552500736	4717415281744513700352	1630888157045153470464	363788805775303516608	49017368477703802176	3546957272430414024	110562887121090768	923444653262076	542578445172	16384	46091647104619717315336

Technical details

Node id: n-e879a52793ead02d · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-dcf8348fea013f35

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:            4,             2;
  2:           36,            36,             4;
  3:          504,           756,           268,             8;
  4:         9576,         19152,         11484,          1908,            16;
  5:       229824,        574560,        486360,        154980,         13420,           32;
  6:      6664896,      19994688,      21881160,      10437840,       1980540,        94068,          64;
  7:    226606464,     793122624,    1065004416,     679704480,     205170840,     24613092,      658732,        128;
  8:   8837652096,   35350608384,   56265052032,   45068026752,   18851744520,   3832487568,   301285692,    4611636,      256;
  9: 388856692224, 1749855115008, 3220984279296, 3107454440832, 1669278590784, 487555643232, 69370288344, 3656933028, 32282476, 512;

%F Formula

The row polynomials satisfy P(n,x) = (5*n-1+(5*n-3)*x)*P(n-1,x) - (5*x+5*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[(5*n - 1 + (5*n - 3)*x)*P[n - 1, x] - (5*x + 5*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] = (5*n - 1 + (5*n - 3)*x)*v[i - 1] - (5*x + 5*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 ?

A257614 → x ↦ 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