A035469 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	1														6
3	40	14	1													55
4	440	208	27	1												676
5	6160	3560	640	44	1											10405
6	104720	70240	15720	1520	65	1										192266
7	2094400	1579760	416080	50680	3080	90	1									4144091
8	48171200	40008640	11981760	1733760	133840	5600	119	1								102034920
9	1252451200	1128404480	375497920	62260800	5748960	307440	9408	152	1							2824680361
10	36321084800	35104585600	12768840000	2367843520	251976480	16201920	636720	14880	189	1						86831184110
11	1162274713600	1194772409600	469245145600	95643363200	11438996800	851447520	40397280	1217040	22440	230	1					2934267713311
12	40679614976000	44174081459200	18556842796800	4103692947200	541764238400	45496897600	2507736000	91512960	2181960	32560	275	1				108104094778956
13	1545825369088000	1763468791884800	786447793331200	186808253632000	26857790960000	2498130835200	155837281600	6625819200	191883120	3712280	45760	324	1			4312068788473485
14	63378840132608000	75611514628249600	35580723905126400	9006010953139200	1395408846832000	141771809379200	9822483070400	473876603200	16028092080	377497120	6046040	62608	377	1		185124583046706226
15	2788668965834752000	3465896998403840000	1712324814264064000	458863238702668800	75985635601075200	8342227506412800	632895962899200	33990189833600	1307337391360	36035439440	703983280	9489480	83720	434	1	8510750110551933315

Technical details

Node id: n-9ae14afeef916271 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-bc765d2be295a474

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           5,           1;
   3:          40,          14,           1;
   4:         440,         208,          27,          1;
   5:        6160,        3560,         640,         44,         1;
   6:      104720,       70240,       15720,       1520,        65,        1;
   7:     2094400,     1579760,      416080,      50680,      3080,       90,      1;
   8:    48171200,    40008640,    11981760,    1733760,    133840,     5600,    119,     1;
   9:  1252451200,  1128404480,   375497920,   62260800,   5748960,   307440,   9408,   152,   1;
  10: 36321084800, 35104585600, 12768840000, 2367843520, 251976480, 16201920, 636720, 14880, 189, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n - 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3*n - 1 + x)*v[i - 1] + x*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.