A166343 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8\,n x + 16\,n x^{2} - 4\,n^{2} x + 25\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 17\,x^{2} - 23\,x^{3} - 2\,n x + 16\,n x^{2} - 29\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 9\,x^{3} - 7\,x^{4} - n x^{2} + 12\,n x^{3} - 11\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,x + n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} + 2\,n x - 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1 + 48\,x + 9\,x^{2}\)

\(P_{2}(x) = 1 + 108\,x + 243\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	1	48	9								58
2	1	108	243	16							368
3	1	232	1458	928	25						2644
4	1	484	6462	11488	3025	36					21496
5	1	992	24831	91904	68975	8928	49				195680
6	1	2012	87885	588336	919275	351540	24647	64			1973760
7	1	4056	295344	3290784	9192750	7404264	1607984	64896	81		21860160
8	1	8148	959040	16838528	76814850	110613384	51567992	6819840	164997	100	263786880

All 15 rows

Technical details

Node id: n-b1948cc23fca675b · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-cde017de88a2ec94

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 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,    48,       9;
   2: 1,   108,     243,        16;
   3: 1,   232,    1458,       928,         25;
   4: 1,   484,    6462,     11488,       3025,          36;
   5: 1,   992,   24831,     91904,      68975,        8928,          49;
   6: 1,  2012,   87885,    588336,     919275,      351540,       24647,         64;
   7: 1,  4056,  295344,   3290784,    9192750,     7404264,     1607984,      64896,         81;
   8: 1,  8148,  959040,  16838528,   76814850,   110613384,    51567992,    6819840,     164997,       100;
   9: 1, 16336, 3042117,  80993792,  568245650,  1327360608,  1113761474,  323975168,   27379053,    408400,    121;
  10: 1, 32716, 9493911, 372649040, 3853650650, 13692079800, 18636441950, 9865345664, 1886535765, 105487900, 989659, 144;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 48*x + 9*x^2, P(2,x) = 1 + 108*x + 243*x^2 + 16*x^3.

%t Mathematica

P[1, x_] := 1 + 48*x + 9*x^2;
P[2, x_] := 1 + 108*x + 243*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 48*x + 9*x^2; v[2] = 1 + 108*x + 243*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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 T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)

References

[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.