A142462 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	18	18	1													37
3	288	432	146	1												867
4	6624	13248	7794	1170	1											28837
5	198720	496800	416160	127440	9362	1										1248483
6	7352640	22057920	23978880	11194560	1995858	74898	1									66654757
7	323516160	1132306560	1513451520	952862400	278214048	30611952	599186	1								4231561827
8	16499324160	65997296640	104686888320	83070126720	34031932704	6610500288	464571954	4793490	1							311365434277
9	956960801280	4306323605760	7906087918080	7575130886400	4011482543040	1138967555040	152864157600	7011720720	38347922	1						26054867535843
10	62202452083200	311012260416000	648670929350400	728610154905600	474155482207680	178582398200640	36545934720960	3475198998720	105520942098	306783378	1					2443360638608677
11	4478576549990400	24632171024947200	57527698568947200	74133360873158400	57293348962417920	26996234468353920	7525344376851840	1139761683878400	78142713045408	1585575181872	2454267026	1				253806227251039587
12	353807547449241600	2122845284695449600	5488184549752819200	7981507639055808000	7147562366853342720	4052501764728468480	1439275824082736640	305105248083018240	34850371232517984	1744508889908928	23805716131314	19634136210	1			28927408930173578917
13	30427449080634777600	197778419024126054400	560731219788448051200	908459099571077683200	925375106177065589760	614088644911792427520	265953224496063836160	73177754614629350400	12029300067964196160	1050784470210926880	38760087036097440	357262449195600	157073089682	1		3589110119708571276003
14	2829752764499034316800	19808269351493240217600	61108931980338110668800	109146090312616541184000	124649977971055778841600	94841148200216158540800	48566667209271424012800	16517482427846883225600	3591727550943737325120	464578336553038868160	31364231637947120640	858438113981273280	5360350395741138	1256584717458	1	481556854136192856053797

Technical details

Node id: n-159b44d7207b424e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d4f1df9ae9c004ef

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,             1;
  2:           18,            18,             1;
  3:          288,           432,           146,             1;
  4:         6624,         13248,          7794,          1170,             1;
  5:       198720,        496800,        416160,        127440,          9362,             1;
  6:      7352640,      22057920,      23978880,      11194560,       1995858,         74898,            1;
  7:    323516160,    1132306560,    1513451520,     952862400,     278214048,      30611952,       599186,          1;
  8:  16499324160,   65997296640,  104686888320,   83070126720,   34031932704,    6610500288,    464571954,    4793490,        1;
  9: 956960801280, 4306323605760, 7906087918080, 7575130886400, 4011482543040, 1138967555040, 152864157600, 7011720720, 38347922, 1;

%F Formula

The row polynomials satisfy P(n,x) = (7*n-5+(7*n-6)*x)*P(n-1,x) - (7*x+7*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[(7*n - 5 + (7*n - 6)*x)*P[n - 1, x] - (7*x + 7*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] = (7*n - 5 + (7*n - 6)*x)*v[i - 1] - (7*x + 7*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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-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) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142462 → 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