A257611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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	48	48	9													105
3	480	720	294	27												1521
4	5760	11520	7392	1632	81											26385
5	80640	201600	177600	64800	8646	243										533529
6	1290240	3870720	4348800	2246400	522768	44688	729									12324345
7	23224320	81285120	111202560	74793600	25445280	4016880	227814	2187								320197761
8	464486400	1857945600	2998517760	2492743680	1128072960	269176320	29940672	1152192	6561							9242042145
9	10218700800	45984153600	85558394880	84861665280	48198689280	15638918400	2721993600	218802240	5800326	19683						293407138089
10	245248819200	1226244096000	2584866816000	2982002688000	2044104007680	845527818240	204692025600	26685964800	1578018288	29119728	59049					10160979432585
11	6376469299200	35070581145600	82618195968000	108752523264000	87487917834240	44190005207040	13857549995520	2571843859200	255953866080	11279085840	145952934	177147				381192465654801
12	178541140377600	1071246842265600	2789460069580800	4127537627136000	3816056298700800	2282714696908800	884570878033920	216153963970560	31361728844160	2416375653120	80121224352	730827552	531441			15400140474054705
13	5356234211328000	34815522373632000	99307132865740800	163218484651622400	170631528741273600	118159363613491200	54728425652060160	16727026594037760	3249065735435520	373975525123200	22548593121600	566695190880	3657326406	1594323		666589878910977849
14	171399494762496000	1199796463337472000	3720618824328806400	6726358922585702400	7849181836777881600	6182768103063552000	3339596421697536000	1231469714777702400	302890450677534720	47473511912386560	4384313893814400	208604290003200	3996124947408	18296197968	4782969	30776150676530816025

Technical details

Node id: n-7599cb43a369a9ba · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-f526dd4eeb36ec75

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:          48,          48,           9;
  3:         480,         720,         294,          27;
  4:        5760,       11520,        7392,        1632,          81;
  5:       80640,      201600,      177600,       64800,        8646,         243;
  6:     1290240,     3870720,     4348800,     2246400,      522768,       44688,        729;
  7:    23224320,    81285120,   111202560,    74793600,    25445280,     4016880,     227814,      2187;
  8:   464486400,  1857945600,  2998517760,  2492743680,  1128072960,   269176320,   29940672,   1152192,    6561;
  9: 10218700800, 45984153600, 85558394880, 84861665280, 48198689280, 15638918400, 2721993600, 218802240, 5800326, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+4+(2*n+1)*x)*P(n-1,x) - (2*x+2*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[(2*n + 4 + (2*n + 1)*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 4 + (2*n + 1)*x)*v[i - 1] - (2*x + 2*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) = 2*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) = 2*n + 3.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257611 → 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