A143491 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 8\,x + 9\,x^{2} - 4\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(8\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(5 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 6 + 20\,x + 9\,x^{2}\)

\(P_{1}(x) = 24 + 104\,x + 81\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	20	9								35
1	24	104	81	16							225
2	120	616	639	224	25						1624
3	720	4176	5220	2480	500	36					13132
4	5040	32112	45936	26640	7375	972	49				118124
5	40320	277056	439740	294784	100625	18396	1715	64			1172700
6	362880	2654784	4581036	3434816	1366225	310464	40474	2816	81		12753576
7	3628800	27999360	51783624	42492224	19029150	5072004	827316	81024	4374	100	150917976

All 15 rows

Technical details

Node id: n-c391da7eb58bddf2 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-505ab518d44a108d

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:         6,         20,          9;
  1:        24,        104,         81,         16;
  2:       120,        616,        639,        224,         25;
  3:       720,       4176,       5220,       2480,        500,         36;
  4:      5040,      32112,      45936,      26640,       7375,        972,        49;
  5:     40320,     277056,     439740,     294784,     100625,      18396,      1715,       64;
  6:    362880,    2654784,    4581036,    3434816,    1366225,     310464,     40474,     2816,      81;
  7:   3628800,   27999360,   51783624,   42492224,   19029150,    5072004,    827316,    81024,    4374,    100;
  8:  39916800,  322508160,  632618424,  559474240,  275714750,   83194020,  16004037,  1971840,  150660,   6500,  121;
  9: 479001600, 4029765120, 8317064448, 7838345856, 4182755500, 1395357480, 305284749, 44565312, 4303530, 264000, 9317, 144;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(4*n+8)*x-9*x^2)*P(n-2,x) + ((4*n+8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 20*x + 9*x^2, P(1,x) = 24 + 104*x + 81*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 6 + 20*x + 9*x^2;
P[1, x_] := 24 + 104*x + 81*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (4*n + 8)*x - 9*x^2)*P[n - 2, x] + ((4*n + 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n + 5 + 4*x)*P[n - 1, x] - 4*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] = 6 + 20*x + 9*x^2; v[2] = 24 + 104*x + 81*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (4*n + 8)*x - 9*x^2)*v[i - 2] + ((4*n + 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n + 5 + 4*x)*v[i - 1] - 4*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 (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
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 Unsigned 2-Stirling numbers of the first kind.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A136124 → (θ+1)²

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.