A156365 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-1 - 3\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 13 + 16\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	13	16	4															33
1	75	134	68	8														285
2	541	1268	984	272	16													3081
3	4683	13586	14248	6384	1072	32												40005
4	47293	163560	216444	135232	39216	4224	64											606033
5	545835	2191934	3505020	2818968	1178384	234912	16704	128										10491885
6	7087261	32410396	60785392	59898448	32907040	9796672	1391872	66304	256									204343641
7	102247563	524566250	1129081088	1318351808	900147296	359905856	79301120	8216576	263936	512								4422082005
8	1622632573	9228981584	22423353396	30284929920	24758721696	12485395968	3779445888	632297472	48503040	1052672	1024							105265315233
9	28091567595	175430841494	474921849652	728574493224	694477837536	423828596160	164295898752	38671029504	5000402688	286797312	4203520	2048						2733583519485
10	526858348381	3583832435844	10697270176584	18377717534480	20023409785200	14364155969664	6812974430976	2086231681536	389129521920	39387591680	1699743744	16797696	4096					76902684021801
11	10641342970443	78315755164994	255521499565080	486045220364688	596084717619280	491913401183712	276539643622656	104785995947520	25847680419072	3874297781760	309816727552	10097627136	67153920	8192				2329889536156005
12	230283190977853	1823124238332088	6455068463192620	13469081526378560	18366868041632336	17149244107221632	11162875329523392	5056140822190080	1561373665324800	314822593316864	38322087191552	2437337464832	60115578880	268533760	16384			75629701786875633
13	5315654681981355	45045837476248814	172016945673978284	390684101456807096	586482228887991664	611481750654317152	452709275672906176	239002174830723456	88994465472463104	22735300815899136	3789382494880768	377624810039296	19194630885376	358544973824	1073954816	32768		2618654297178083085
14	130370767029135901	1177442182762911212	4823780353114456800	11846963998282795600	19416504691253223680	22365989351052071616	18570222425673505792	11210255887647100160	4898371628862097920	1522429427505996800	325508485996797952	45242254367944704	3714003123077120	151390741872640	2141603758080	4295426048	65536	96336948993312237561

Technical details

Node id: n-984f75403d58b29b · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4358062586543316

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:          13,           16,            4;
  1:          75,          134,           68,            8;
  2:         541,         1268,          984,          272,           16;
  3:        4683,        13586,        14248,         6384,         1072,           32;
  4:       47293,       163560,       216444,       135232,        39216,         4224,           64;
  5:      545835,      2191934,      3505020,      2818968,      1178384,       234912,        16704,         128;
  6:     7087261,     32410396,     60785392,     59898448,     32907040,      9796672,      1391872,       66304,        256;
  7:   102247563,    524566250,   1129081088,   1318351808,    900147296,    359905856,     79301120,     8216576,     263936,       512;
  8:  1622632573,   9228981584,  22423353396,  30284929920,  24758721696,  12485395968,   3779445888,   632297472,   48503040,   1052672,    1024;
  9: 28091567595, 175430841494, 474921849652, 728574493224, 694477837536, 423828596160, 164295898752, 38671029504, 5000402688, 286797312, 4203520, 2048;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+5+(2*n+4)*x)*P(n-1,x) - (1+3*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 16*x + 4*x^2.

%t Mathematica

P[0, x_] := 13 + 16*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(2*n + 5 + (2*n + 4)*x)*P[n - 1, x] - (1 + 3*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] = 13 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 5 + (2*n + 4)*x)*v[i - 1] - (1 + 3*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 T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) =…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A142075 → 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