A225476 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−3, −1)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(3 + 4\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

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	7	8	2													17
3	38	67	36	6												147
4	277	656	548	192	24											1697
5	2522	7501	8520	4620	1200	120										24483
6	27547	98648	141782	104640	41880	8640	720									423857
7	351038	1470127	2556876	2392866	1302000	412440	70560	5040								8560947
8	5112457	24513056	50044808	56794752	39194064	16853760	4415040	645120	40320							197613377
9	83764082	452459641	1060626960	1415589720	1185478560	646223760	229461120	51226560	6531840	362880						5131725123
10	1524907087	9162283688	24251496362	37225879680	36694329360	24274615680	10919089440	3299788800	641692800	72576000	3628800					148070287697
11	30536665238	202007587387	595872448116	1034248756926	1173520682400	914056801680	498839584320	190827917280	50175417600	8641987200	878169600	39916800				4699645934547
12	667096092637	4817816878256	15670978468268	30349348130112	38971219336104	34955965566720	22461556265280	10420924631040	3465921191040	806319360000	124620249600	11496038400	479001600			162723741209057
13	15787642820042	123599868171781	439489551610200	939558441259620	1347218824071120	1368195351923400	1012575673549440	553040741373120	222979340263680	65572033887360	13676613350400	1916884569600	161902540800	6227020800		6103779096411363
14	402374890155427	3394324161511928	13098121223782142	30638912721318720	48531102877169640	55062753986560320	46149995192661360	29030756122368000	13776068414908800	4908849750604800	1293395290387200	244447928524800	31340595686400	2440992153600	87178291200	246564971326084337

Technical details

Node id: n-4e118a5f86e56b36 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-470e0be37b3b9576

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:        7,         8,          2;
  3:       38,        67,         36,          6;
  4:      277,       656,        548,        192,         24;
  5:     2522,      7501,       8520,       4620,       1200,       120;
  6:    27547,     98648,     141782,     104640,      41880,      8640,       720;
  7:   351038,   1470127,    2556876,    2392866,    1302000,    412440,     70560,     5040;
  8:  5112457,  24513056,   50044808,   56794752,   39194064,  16853760,   4415040,   645120,   40320;
  9: 83764082, 452459641, 1060626960, 1415589720, 1185478560, 646223760, 229461120, 51226560, 6531840, 362880;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (3+4*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 + x)*P[n - 1, x] + (3 + 4*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 + x)*v[i - 1] + (3 + 4*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 (-3, -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, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…?
  - 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