A049029 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-42 + 14\,x - x^{2} + 52\,n - 8\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(7\,x - x^{2} - 4\,n x\right)\,P_{n-2}'(x) + \left(-9 + 2\,x + 8\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	15	2													17
4	255	60	3												318
5	5175	1650	150	4											6979
6	123795	48300	6075	300	5										178475
7	3427515	1551690	242550	16800	525	6									5239086
8	108046575	54957420	10089135	877800	38850	840	7								174010627
9	3824996175	2139602850	445644990	45930780	2573550	79380	1260	8							6458828993
10	150346471275	91094503500	21035203875	2477355300	165502575	6501600	148050	1800	9						265125687984
11	6499426608675	4217756593050	1062190725750	139527927000	10709812575	504178290	14691600	257400	2475	10					11930130796825
12	306553491419175	211233413090700	57311789725575	8253122724000	709900537500	38564867880	1352171205	30432600	423225	3300	11				584101665395171
13	15668768604864375	11386011050464050	3297063185365950	513831224006100	48651032430000	2972948378400	120713933340	3279996720	58783725	664950	4290	12			30917422098891912
14	862827112324051875	657568144985251500	201714554956189275	33684464015502300	3464048910947625	233785193241600	10711275775200	338038580880	7334587260	107207100	1006005	5460	13		1759503167142346093
15	50929793720847916875	40522174778777942250	13089225514849233750	2323705044887226000	256876612498130625	18885325867357950	958271041728000	34213965786000	864376162650	15332417100	186336150	1474200	6825	14	107141654362631718389

Technical details

Node id: n-d1d9cf047e7ff48c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-fd383b95e7b61cc6

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:            0;
   2:            1;
   3:           15,           2;
   4:          255,          60,           3;
   5:         5175,        1650,         150,          4;
   6:       123795,       48300,        6075,        300,         5;
   7:      3427515,     1551690,      242550,      16800,       525,       6;
   8:    108046575,    54957420,    10089135,     877800,     38850,     840,      7;
   9:   3824996175,  2139602850,   445644990,   45930780,   2573550,   79380,   1260,    8;
  10: 150346471275, 91094503500, 21035203875, 2477355300, 165502575, 6501600, 148050, 1800, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-52*n+42+(8*n-14)*x+x^2)*P(n-2,x) - ((4*n-7)*x+x^2)*d/dx P(n-2,x) + (8*n-9+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 52*n + 42 + (8*n - 14)*x + x^2)*P[n - 2, x] - ((4*n - 7)*x + x^2)*D[P[n - 2, x], x] + (8*n - 9 + 2*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 52*n + 42 + (8*n - 14)*x + x^2)*v[i - 2] - ((4*n - 7)*x + x^2)*deriv(v[i - 2], x) + (8*n - 9 + 2*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 quartic factorial numbers A007696(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.