A209414 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(3\,x + 3\,x^{3} - 12\,x^{4} + 2\,n x + 15\,n x^{2} + 152\,n x^{3} + 88\,n x^{4} + n^{2} - 9\,n^{2} x + 61\,n^{2} x^{2} - 114\,n^{2} x^{3} + 9\,n^{3} x - 44\,n^{3} x^{2} + 80\,n^{3} x^{3} - 128\,n^{3} x^{4} + 9\,n^{4} x^{2} - 48\,n^{4} x^{3} + 64\,n^{4} x^{4}\right)\,P_{n}(x) = \left(x + 4\,x^{2} + 36\,x^{3} + 32\,x^{4} - 61\,x^{5} - 12\,x^{6} + 2\,n x + 9\,n x^{2} + 30\,n x^{3} - 69\,n x^{4} - 12\,n x^{5} + 40\,n x^{6} + n^{2} x + 17\,n^{2} x^{2} - 35\,n^{2} x^{3} - 145\,n^{2} x^{4} + 162\,n^{2} x^{5} + 9\,n^{3} x^{2} - 17\,n^{3} x^{3} - 104\,n^{3} x^{4} + 240\,n^{3} x^{5} - 128\,n^{3} x^{6} + 9\,n^{4} x^{3} - 57\,n^{4} x^{4} + 112\,n^{4} x^{5} - 64\,n^{4} x^{6}\right)\,P_{n-2}(x) + \left(2\,x + 44\,x^{2} + 210\,x^{3} + 676\,x^{4} + 256\,x^{5} + 54\,n x^{2} + 272\,n x^{3} + 292\,n x^{4} + 48\,n x^{5} + n^{2} + 2\,n^{2} x + 17\,n^{2} x^{2} + 68\,n^{2} x^{3} - 436\,n^{2} x^{4} - 384\,n^{2} x^{5} + 9\,n^{3} x - 8\,n^{3} x^{2} - 68\,n^{3} x^{3} - 32\,n^{3} x^{4} + 9\,n^{4} x^{2} - 30\,n^{4} x^{3} - 32\,n^{4} x^{4} + 128\,n^{4} x^{5}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	16	9						26
3	1	28	81	16					126
4	1	40	234	256	25				556
5	1	52	468	1120	625	36			2302
6	1	64	783	3040	3875	1296	49		9108
7	1	76	1179	6448	13825	10836	2401	64	34830

All 15 rows

Technical details

Node id: n-57ebbee3d0c93bf9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9974c1fe8b286c8c

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,  16,    9;
  3: 1,  28,   81,    16;
  4: 1,  40,  234,   256,    25;
  5: 1,  52,  468,  1120,   625,     36;
  6: 1,  64,  783,  3040,  3875,   1296,     49;
  7: 1,  76, 1179,  6448, 13825,  10836,   2401,    64;
  8: 1,  88, 1656, 11776, 36550,  49392,  26068,  4096,   81;
  9: 1, 100, 2214, 19456, 80150, 160056, 148176, 56064, 6561, 100;

%F Formula

The row polynomials satisfy (n^2+(9*n^3-9*n^2+2*n+3)*x+(9*n^4-44*n^3+61*n^2+15*n)*x^2-(48*n^4-80*n^3+114*n^2-152*n-3)*x^3+(64*n^4-128*n^3+88*n-12)*x^4)*P(n,x) = ((n^2+2*n+1)*x+(9*n^3+17*n^2+9*n+4)*x^2+(9*n^4-17*n^3-35*n^2+30*n+36)*x^3-(57*n^4+104*n^3+145*n^2+69*n-32)*x^4+(112*n^4+240*n^3+162*n^2-12*n-61)*x^5-(64*n^4+128*n^3-40*n+12)*x^6)*P(n-2,x) + (n^2+(9*n^3+2*n^2+2)*x+(9*n^4-8*n^3+17*n^2+54*n+44)*x^2-(30*n^4+68*n^3-68*n^2-272*n-210)*x^3-(32*n^4+32*n^3+436*n^2-292*n-676)*x^4+(128*n^4-384*n^2+48*n+256)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 + 2*n + 1)*x + (9*n^3 + 17*n^2 + 9*n + 4)*x^2 + (9*n^4 - 17*n^3 - 35*n^2 + 30*n + 36)*x^3 - (57*n^4 + 104*n^3 + 145*n^2 + 69*n - 32)*x^4 + (112*n^4 + 240*n^3 + 162*n^2 - 12*n - 61)*x^5 - (64*n^4 + 128*n^3 - 40*n + 12)*x^6)*P[n - 2, x] + (n^2 + (9*n^3 + 2*n^2 + 2)*x + (9*n^4 - 8*n^3 + 17*n^2 + 54*n + 44)*x^2 - (30*n^4 + 68*n^3 - 68*n^2 - 272*n - 210)*x^3 - (32*n^4 + 32*n^3 + 436*n^2 - 292*n - 676)*x^4 + (128*n^4 - 384*n^2 + 48*n + 256)*x^5)*P[n - 1, x])/(n^2 + (9*n^3 - 9*n^2 + 2*n + 3)*x + (9*n^4 - 44*n^3 + 61*n^2 + 15*n)*x^2 - (48*n^4 - 80*n^3 + 114*n^2 - 152*n - 3)*x^3 + (64*n^4 - 128*n^3 + 88*n - 12)*x^4)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + 2*n + 1)*x + (9*n^3 + 17*n^2 + 9*n + 4)*x^2 + (9*n^4 - 17*n^3 - 35*n^2 + 30*n + 36)*x^3 - (57*n^4 + 104*n^3 + 145*n^2 + 69*n - 32)*x^4 + (112*n^4 + 240*n^3 + 162*n^2 - 12*n - 61)*x^5 - (64*n^4 + 128*n^3 - 40*n + 12)*x^6)*v[i - 2] + (n^2 + (9*n^3 + 2*n^2 + 2)*x + (9*n^4 - 8*n^3 + 17*n^2 + 54*n + 44)*x^2 - (30*n^4 + 68*n^3 - 68*n^2 - 272*n - 210)*x^3 - (32*n^4 + 32*n^3 + 436*n^2 - 292*n - 676)*x^4 + (128*n^4 - 384*n^2 + 48*n + 256)*x^5)*v[i - 1])/(n^2 + (9*n^3 - 9*n^2 + 2*n + 3)*x + (9*n^4 - 44*n^3 + 61*n^2 + 15*n)*x^2 - (48*n^4 - 80*n^3 + 114*n^2 - 152*n - 3)*x^3 + (64*n^4 - 128*n^3 + 88*n - 12)*x^4)); 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 of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.