A142462 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

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

Technical details

Node id: n-58a84645df5825fb · revision 0 · JSON
Definition source: derived:mobius_x_over_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: 1,        2;
  2: 1,       18,         18;
  3: 1,      146,        432,          288;
  4: 1,     1170,       7794,        13248,          6624;
  5: 1,     9362,     127440,       416160,        496800,        198720;
  6: 1,    74898,    1995858,     11194560,      23978880,      22057920,       7352640;
  7: 1,   599186,   30611952,    278214048,     952862400,    1513451520,    1132306560,     323516160;
  8: 1,  4793490,  464571954,   6610500288,   34031932704,   83070126720,  104686888320,   65997296640,   16499324160;
  9: 1, 38347922, 7011720720, 152864157600, 1138967555040, 4011482543040, 7575130886400, 7906087918080, 4306323605760, 956960801280;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*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[((1 + 4*x + 5*x^2 + 2*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] = ((1 + 4*x + 5*x^2 + 2*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) (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+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