A144436 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(152\,x^{2}\right)\,P_{n-4}(x) + \left(68\,x^{3}\right)\,P_{n-4}'(x) + \left(-16\,x - 48\,x^{2} + 74\,n x^{2}\right)\,P_{n-3}(x) + \left(17\,x^{2} - 77\,x^{3} + 33\,n x^{3}\right)\,P_{n-3}'(x) + \left(17\,x^{3} - 17\,x^{4}\right)\,P_{n-3}''(x) + \left(20\,x - 7\,x^{2} - 4\,n x - 2\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 8\,x^{2} - 13\,x^{3} + 4\,n x^{2} - 14\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3} - 3\,x^{4} + 4\,n x^{3} - 4\,n x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

\(P_{2}(x) = 1 + 32\,x + 9\,x^{2}\)

\(P_{3}(x) = 1 + 92\,x + 207\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	32	9						42
3	1	92	207	16					316
4	1	216	1530	864	25				2636
5	1	468	7362	13088	2925	36			23880
6	1	976	29295	115584	81375	8784	49		236064
7	1	1996	105273	775088	1211075	421092	24451	64	2539040

All 15 rows

Technical details

Node id: n-bb5bc35b3e1328a3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-1009aec5f5b483bb

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 0)
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: 1;
  1: 1,    4;
  2: 1,   32,       9;
  3: 1,   92,     207,       16;
  4: 1,  216,    1530,      864,        25;
  5: 1,  468,    7362,    13088,      2925,        36;
  6: 1,  976,   29295,   115584,     81375,      8784,       49;
  7: 1, 1996,  105273,   775088,   1211075,    421092,    24451,      64;
  8: 1, 4040,  356040,  4431584,  12833150,   9971064,  1938440,   64640,     81;
  9: 1, 8132, 1158804, 22905664, 110556150, 159200856, 70148596, 8240384, 164673, 100;

%F Formula

The row polynomials satisfy P(n,x) = 152*x^2*P(n-4,x) + 68*x^3*d/dx P(n-4,x) - (16*x-(74*n-48)*x^2)*P(n-3,x) + (17*x^2+(33*n-77)*x^3)*d/dx P(n-3,x) + (17*x^3-17*x^4)*d^2/dx^2 P(n-3,x) - ((4*n-20)*x-(9*n^2-2*n-7)*x^2)*P(n-2,x) - (2*x-(4*n-8)*x^2-(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) - (x^2-(4*n+4)*x^3+(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (2*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 32*x + 9*x^2, P(3,x) = 1 + 92*x + 207*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 32*x + 9*x^2;
P[3, x_] := 1 + 92*x + 207*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[152*x^2*P[n - 4, x] + 68*x^3*D[P[n - 4, x], x] - (16*x - (74*n - 48)*x^2)*P[n - 3, x] + (17*x^2 + (33*n - 77)*x^3)*D[P[n - 3, x], x] + (17*x^3 - 17*x^4)*D[P[n - 3, x], {x, 2}] - ((4*n - 20)*x - (9*n^2 - 2*n - 7)*x^2)*P[n - 2, x] - (2*x - (4*n - 8)*x^2 - (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 4*n*x)*P[n - 1, x] + (2*x - (4*n + 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] = 1; v[2] = 1 + 4*x; v[3] = 1 + 32*x + 9*x^2; v[4] = 1 + 92*x + 207*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = 152*x^2*v[i - 4] + 68*x^3*deriv(v[i - 4], x) - (16*x - (74*n - 48)*x^2)*v[i - 3] + (17*x^2 + (33*n - 77)*x^3)*deriv(v[i - 3], x) + (17*x^3 - 17*x^4)*deriv(deriv(v[i - 3], x), x) - ((4*n - 20)*x - (9*n^2 - 2*n - 7)*x^2)*v[i - 2] - (2*x - (4*n - 8)*x^2 - (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 4*n*x)*v[i - 1] + (2*x - (4*n + 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 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.