A257623 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	66	66	9													141
3	1056	1584	582	27												3249
4	22176	44352	26994	4818	81											98421
5	576576	1441440	1230240	403920	39030	243										3691449
6	17873856	53621568	58988160	28607040	5680290	313698	729									165085341
7	643458816	2252105856	3035096064	1957475520	605811360	77294448	2513958	2187								8573758209
8	26381811456	105527245824	168407471232	135877053312	57743975520	12141315648	1032481362	20124786	6561							507131485701
9	1213563326976	5461034971392	10072704305664	9769635203328	5305900856256	1583079065568	235063383456	13643630352	161037654	19683						33654785800329
10	61891729675776	309458648378880	647805380302080	734469630935040	486880888253760	189725279679936	41347149560640	4449438987840	179138608770	1288419330	59049					2476208572861101
11	3465936861843456	19062652740139008	44653062616081920	57968886221326080	45437557952666880	21948552982452096	6387333719614848	1044422658639360	82956119521440	2342974526640	10307708934	177147				200053715154697809
12	211422148572450816	1268532891434704896	3287626788647545344	4809915771752931840	4355601983576363520	2516775109569616896	922309593728979456	206000333273839104	25763799658655520	1530697743812160	30572053644594	82462734354	531441			17605509772475809941
13	13953861805781753856	90700101737581400064	257682007867475423232	419549924157719427072	431233648288653450240	290478664786137924096	128840599537888702464	36822226924545334272	6437868122792538432	624712044769132320	28041712246932384	398343787457616	659705063478	1594323		1676352055989086133849
14	990724188210504523776	6935069317473531666432	21432708889383629021184	38440352209145862463488	44229062043080455597056	34061233698017710119936	17773047166857367537152	6221542951723402727424	1414707400316564722368	196502820602188516416	14957252986875623424	511124321044104768	5185725992647266	5277650073762	4782969	171710424253122784127421

Technical details

Node id: n-f36340a6b18e36e9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-04577fbe25f76a46

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:             6,             3;
  2:            66,            66,              9;
  3:          1056,          1584,            582,            27;
  4:         22176,         44352,          26994,          4818,            81;
  5:        576576,       1441440,        1230240,        403920,         39030,           243;
  6:      17873856,      53621568,       58988160,      28607040,       5680290,        313698,          729;
  7:     643458816,    2252105856,     3035096064,    1957475520,     605811360,      77294448,      2513958,        2187;
  8:   26381811456,  105527245824,   168407471232,  135877053312,   57743975520,   12141315648,   1032481362,    20124786,      6561;
  9: 1213563326976, 5461034971392, 10072704305664, 9769635203328, 5305900856256, 1583079065568, 235063383456, 13643630352, 161037654, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (5*n+1+(5*n-2)*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 - 2)*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 - 2)*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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
  - 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257623 → 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