A049029 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-56 + 16\,x - x^{2} + 60\,n - 8\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(8\,x - x^{2} - 4\,n x\right)\,P_{n-2}'(x) + \left(-11 + 2\,x + 8\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	5	2							7
3	45	30	3						78
4	585	510	90	4					1189
5	9945	10350	2475	200	5				22975
6	208845	247590	72450	8100	375	6			537366
7	5221125	6855030	2327535	323400	21000	630	7		14748727
8	151412625	216093150	82436130	13452180	1097250	46620	980	8	464538943

All 15 rows

Technical details

Node id: n-efcf89935f283045 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-81bb74537eeb9ad3

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,            2;
   3:           45,           30,            3;
   4:          585,          510,           90,           4;
   5:         9945,        10350,         2475,         200,          5;
   6:       208845,       247590,        72450,        8100,        375,         6;
   7:      5221125,      6855030,      2327535,      323400,      21000,       630,       7;
   8:    151412625,    216093150,     82436130,    13452180,    1097250,     46620,     980,      8;
   9:   4996616625,   7649992350,   3209404275,   594193320,   57413475,   3088260,   92610,   1440,    9;
  10: 184874815125, 300692942550, 136641755250, 28046938500, 3096694125, 198603090, 7585200, 169200, 2025, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-60*n+56+(8*n-16)*x+x^2)*P(n-2,x) - ((4*n-8)*x+x^2)*d/dx P(n-2,x) + (8*n-11+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*P[n - 2, x] - ((4*n - 8)*x + x^2)*D[P[n - 2, x], x] + (8*n - 11 + 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] = 1; v[2] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*v[i - 2] - ((4*n - 8)*x + x^2)*deriv(v[i - 2], x) + (8*n - 11 + 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.