A257627 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	78	78	9													165
3	1560	2340	834	27												4761
4	42120	84240	50622	8502	81											185565
5	1432080	3580200	3034200	971100	85506	243										9103329
6	58715280	176145840	192909600	92242800	17620278	856518	729									538491045
7	2818333440	9864167040	13252468320	8470753200	2566232760	310679460	8569554	2187								37291205961
8	155008339200	620033356800	987238717920	791599404960	331881633720	67803175440	5392955022	85708662	6561							2959043298285
9	9610517030400	43247326636800	79629193013760	76881317909760	41364752934960	12119016382200	1735135311000	92794447980	857125986	19683						264680910812529
10	663125675097600	3315628375488000	6931197887328000	7831021296384000	5155614176272560	1983907168523280	422538161245200	43499136423600	1588648253478	8571377958	59049					26348129096452725
11	50397551307417600	277186532190796800	648565741796544000	839646846653472000	654682587873050880	313342831841719680	89754769886221440	14273159682038400	1075752239235960	27118448222580	85714133874	177147				2888952977633030361
12	4182996758515660800	25097980551093964800	64985653033889702400	94863443451087168000	85546706285572646400	49083950497801190400	17780372998212464640	3895968684865720320	471513260602561320	26360422706114640	462127903524222	857142401622	531441			345935408929393651005
13	376469708266409472000	2447053103731661568000	6946748074663134566400	11289530269598962867200	11564883057477257164800	7746872948227851609600	3405823329799860203520	959851500999055441920	164078260885384457040	15328642491744496200	641892703017235800	7867317211132860	8571427204866	1594323		44917288664732979014529
14	36517561701841718784000	255622931912892031488000	789473710221694725427200	1413744146462571943219200	1622112640646655033676800	1243588404583625772672000	644385787084435556160000	223175685197547391795200	49920526241258257625040	6756147758363922238320	492519020225544748800	15562771216636316400	133855821142921878	85714281614598	4782969	6285805757543863963470405

Technical details

Node id: n-b9fa97756c104cb4 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-db62b20fb03c4b11

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:            78,             78,              9;
  3:          1560,           2340,            834,             27;
  4:         42120,          84240,          50622,           8502,             81;
  5:       1432080,        3580200,        3034200,         971100,          85506,            243;
  6:      58715280,      176145840,      192909600,       92242800,       17620278,         856518,           729;
  7:    2818333440,     9864167040,    13252468320,     8470753200,     2566232760,      310679460,       8569554,        2187;
  8:  155008339200,   620033356800,   987238717920,   791599404960,   331881633720,    67803175440,    5392955022,    85708662,      6561;
  9: 9610517030400, 43247326636800, 79629193013760, 76881317909760, 41364752934960, 12119016382200, 1735135311000, 92794447980, 857125986, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (7*n-1+(7*n-4)*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 - 1 + (7*n - 4)*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 - 1 + (7*n - 4)*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) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 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(x) = 7*x + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257627 → 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