A094262 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 4\,x + x^{2}\right)\,P_{n-1}(x) + \left(4 + 8\,x + 5\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	4	1													9
2	32	72	60	22	3											189
3	416	1472	2144	1648	706	160	15									6561
4	7552	36480	76320	90400	66360	30936	8950	1470	105							318573
5	176128	1078528	2947072	4734400	4954240	3530224	1735504	581476	127120	16380	945					19882017
6	5018624	37223424	125637120	255247104	347751936	334821312	233675232	119144880	44059932	11527768	2026080	214830	10395			1516358637
7	168968192	1471852544	5915737088	14544316416	24442102272	29707378176	26935746432	18512496384	9693451968	3848949616	1141043260	244949320	36001350	3243240	135135	136666371393

All 15 rows

Technical details

Node id: n-5a364d1118d1dc6e · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-982865a374cd90cf

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:            4,             4,              1;
  2:           32,            72,             60,             22,               3;
  3:          416,          1472,           2144,           1648,             706,             160,              15;
  4:         7552,         36480,          76320,          90400,           66360,           30936,            8950,            1470,             105;
  5:       176128,       1078528,        2947072,        4734400,         4954240,         3530224,         1735504,          581476,          127120,           16380,            945;
  6:      5018624,      37223424,      125637120,      255247104,       347751936,       334821312,       233675232,       119144880,        44059932,        11527768,        2026080,         214830,          10395;
  7:    168968192,    1471852544,     5915737088,    14544316416,     24442102272,     29707378176,     26935746432,     18512496384,      9693451968,      3848949616,     1141043260,      244949320,       36001350,       3243240,       135135;
  8:   6563282944,   65664000000,   306262179840,   884078520320,   1768152468480,   2598370274304,   2902733191680,   2514999893760,   1708254504000,    912773758880,   382465002816,   124369234440,    30772269220,    5601315720,    707476770,     55405350,     2027025;
  9: 288909131776, 3264318570496, 17331725123584, 57463636582400, 133337172776960, 229965861994496, 305555623289856, 319847287916544, 267425289790080, 179952603113600, 97699511273024, 42699886231904, 14913395202376, 4106416840240, 871921910860, 137771694060, 15256741500, 1056755700, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (4+4*x+x^2)*P(n-1,x) + (4+8*x+5*x^2+x^3)*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[(4 + 4*x + x^2)*P[n - 1, x] + (4 + 8*x + 5*x^2 + x^3)*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] = (4 + 4*x + x^2)*v[i - 1] + (4 + 8*x + 5*x^2 + x^3)*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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.