A112486 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in [−2, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	10	11	3													24
3	82	141	80	15												318
4	938	2199	1915	735	105											5892
5	13778	40937	48244	28210	8190	945										140304
6	247210	889699	1324211	1043882	459900	107415	10395									4082712
7	5240338	22152197	39865104	39606385	23470370	8298675	1621620	135135								140389824
8	128149802	622274863	1314040371	1576573625	1175827345	558362805	164909745	27702675	2027025							5569868256
9	3551246162	19477281825	47219159284	66428550716	59777331900	35690644990	14141667540	3586482900	528377850	34459425						250435202592
10	109979486890	672366580731	1840519545755	2971276537372	3133352230008	2255755231730	1122953801970	381766525140	84839104350	11130394275	654729075					12584594167296
11	3764281873042	25381017057085	77430605474144	141103144593411	170689399660780	143938672647770	86354967138440	36863600884110	10974743349570	2170426883625	256653797400	13749310575				698941262669952
12	141104799067178	1040181264914375	3499490253526891	7105947445506819	9700752283166945	9380898233496790	6589956467204110	3388847341869990	1266263781482700	335319714004275	59740754448375	6430094245575	316234143225			42515248667077248
13	5749087305575378	45997580084497753	169181910494702676	378806532660977590	576132728223420914	628609093993655775	506258115546102920	304745218448490820	137125890763611750	45558299342330775	10863978030605700	1761002532238950	173928778773750	7905853580625		2810971272058565376
14	252969604725106090	2183139846876296339	8714987854053222819	21331814102158780958	35771111124110013340	43475677536302009817	39497879159279065525	27251692267115896580	14350245295288072170	5739755621239765125	1716693666357995325	372330917891682750	55362057340590000	5051840438019375	213458046676875	200718924351223193088

Technical details

Node id: n-6e68ecd44a6b5992 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-010a81a2cff13557

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:          2,           1;
  2:         10,          11,           3;
  3:         82,         141,          80,          15;
  4:        938,        2199,        1915,         735,         105;
  5:      13778,       40937,       48244,       28210,        8190,         945;
  6:     247210,      889699,     1324211,     1043882,      459900,      107415,       10395;
  7:    5240338,    22152197,    39865104,    39606385,    23470370,     8298675,     1621620,     135135;
  8:  128149802,   622274863,  1314040371,  1576573625,  1175827345,   558362805,   164909745,   27702675,   2027025;
  9: 3551246162, 19477281825, 47219159284, 66428550716, 59777331900, 35690644990, 14141667540, 3586482900, 528377850, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+n*x)*P(n-1,x) + (2+3*x+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 + n*x)*P[n - 1, x] + (2 + 3*x + 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 + n*x)*v[i - 1] + (2 + 3*x + 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 [-2, -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 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.?
  - 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