A225473 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 21\,x + 49\,x^{2} + 47\,x^{3} + 16\,x^{4} - 4\,n x - 16\,n x^{2} - 20\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 20\,x^{2} + 36\,x^{3} + 28\,x^{4} + 8\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	7														10
3	9	58	81													148
4	27	397	1385	1399												3208
5	81	2644	17366	40948	32289											93328
6	243	17779	196366	811254	1441231	931687										3398560
7	729	121294	2132527	13759908	41329447	58891822	32259441									148495168
8	2187	836665	22816335	216474549	978456881	2305486971	2742082101	1303134679								7569290368
9	6561	5810728	243327228	3275407320	20925145574	72517350552	140467243452	143355570664	60160827969							440950690048
10	19683	40511071	2600640140	48562381804	421927539482	2018161118098	5661118184156	9305566719580	8316131752859	3124574220487						28898683087360
11	59049	283006690	27916936157	712839817816	8210924074946	52103334639820	198536084053970	467319494446552	667003944507221	530150154527266	180312309395601					2104377285465088
12	177147	1979098213	301138433317	10420968360147	156359392963374	1281621528042354	6373139949134922	20145678220355142	40825277584799847	51481436663801049	36842627781440257	11445969681199159				168562834887804928
13	531441	13847133052	3263029722226	152241397912044	2938725530745135	30530306492521848	192795947765668092	786058089704698680	2120805993859289439	3771805704340067340	4259608335696609202	2772195694255085020	792626097462398049			14729520413382381568
14	1594323	96908142283	35491709067642	2226930915694242	54813535716819265	711806747393551281	5597487979063264956	28634777998000146444	98774502149625105189	232302428543335598205	368008348822241445418	376278919245859506738	224540005776649290087	59462922484586318887		1394368275802005544960
15	4782969	678285251446	387210665239131	32649334663333980	1018291529980365361	16357841323946597898	157862702074000435947	993254661539525411592	4260404590331341555395	12722824978916961665322	26508431911601520632329	37852057652836766080476	35355605897265096190611	19479458600606775149590	4804064349575887075761	142151374514825419767808

Technical details

Node id: n-7cfc47df0195bedb · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-0833365c0b4ddd0e

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,        7;
   3:     9,       58,         81;
   4:    27,      397,       1385,        1399;
   5:    81,     2644,      17366,       40948,        32289;
   6:   243,    17779,     196366,      811254,      1441231,        931687;
   7:   729,   121294,    2132527,    13759908,     41329447,      58891822,      32259441;
   8:  2187,   836665,   22816335,   216474549,    978456881,    2305486971,    2742082101,    1303134679;
   9:  6561,  5810728,  243327228,  3275407320,  20925145574,   72517350552,  140467243452,  143355570664,   60160827969;
  10: 19683, 40511071, 2600640140, 48562381804, 421927539482, 2018161118098, 5661118184156, 9305566719580, 8316131752859, 3124574220487;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3-(4*n-21)*x-(16*n-49)*x^2-(20*n-47)*x^3-(8*n-16)*x^4)*P(n-1,x) + (4*x+20*x^2+36*x^3+28*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*P[n - 1, x] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3 - (4*n - 21)*x - (16*n - 49)*x^2 - (20*n - 47)*x^3 - (8*n - 16)*x^4)*v[i - 1] + (4*x + 20*x^2 + 36*x^3 + 28*x^4 + 8*x^5)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741