A257624 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	10	5														15
2	130	130	25													285
3	2080	3120	1290	125												6615
4	39520	79040	51090	11570	625											181845
5	869440	2173600	1924000	712400	98810	3125										5781375
6	21736000	65208000	73507200	38334400	9120930	821730	15625									208743885
7	608608000	2130128000	2921318400	1977976000	682616480	111012720	6730090	78125								8438467815
8	18866848000	75467392000	122025904000	101941840000	46595305120	11332834240	1308631090	54621970	390625							377593767045
9	641472832000	2886627744000	5378877504000	5355141792000	3063933512640	1007444037600	179597776800	15105027600	440882010	1953125						18528643061775
10	23734494784000	118672473920000	250457407200000	289794785280000	199766599032000	83442439080000	20538906398400	2756049316800	171886769730	3546587330	9765625					989338598133885
11	949379791360000	5221588852480000	12313071010240000	16246903152480000	13128129190176000	6680692754736000	2120704722364800	401526345792000	41334878750880	1936860102320	28470354890	48828125				57105296077665015
12	40823331028480000	244939986170880000	638339237215680000	946412979512000000	878132196317376000	528390756077184000	206673551418048000	51247674054720000	7611310574730720	609678064149440	21675575739090	228251120370	244140625			3543202604504248245
13	1877873227310080000	12206175977515520000	34841288963120640000	57359153544492800000	60141710019174784000	41845947131596608000	19525019097712128000	6034683533983104000	1192402313738475840	140976162989261600	8882302109917600	241398597694800	1828450369210	1220703125		235174355502012086175
14	92015788138193920000	644110516967357440000	1998628640492323840000	3618335122378296320000	4232813644766715008000	3347296923087877760000	1818808058845377600000	676621635492042240000	168607779125073288000	26949521860533272000	2565358510905881600	128214607101963200	2679154429442530	14639809984930	6103515625	16626883898072141475885

Technical details

Node id: n-074080256a21a584 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-1d32ce7fdc01b2c4

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:           10,             5;
  2:          130,           130,            25;
  3:         2080,          3120,          1290,           125;
  4:        39520,         79040,         51090,         11570,           625;
  5:       869440,       2173600,       1924000,        712400,         98810,          3125;
  6:     21736000,      65208000,      73507200,      38334400,       9120930,        821730,        15625;
  7:    608608000,    2130128000,    2921318400,    1977976000,     682616480,     111012720,      6730090,       78125;
  8:  18866848000,   75467392000,  122025904000,  101941840000,   46595305120,   11332834240,   1308631090,    54621970,    390625;
  9: 641472832000, 2886627744000, 5378877504000, 5355141792000, 3063933512640, 1007444037600, 179597776800, 15105027600, 440882010, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (3*n+7+(3*n+2)*x)*P(n-1,x) - (3*x+3*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[(3*n + 7 + (3*n + 2)*x)*P[n - 1, x] - (3*x + 3*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] = (3*n + 7 + (3*n + 2)*x)*v[i - 1] - (3*x + 3*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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257624 → 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