A166972 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-8\,x - 7\,n x^{2} + 8\,n^{2} x - 2\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(16\,x - 10\,x^{2} + 16\,n x + 5\,n x^{2} - 13\,n x^{3} + 19\,n^{2} x^{2} - 14\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(26\,x^{2} - 2\,x^{3} + 26\,n x^{2} + 31\,n x^{3} - 3\,n x^{4} + 43\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(6\,x^{3} + 6\,n x^{3} + 9\,n x^{4} + 12\,n^{2} x^{4}\right)\,P_{n-2}'''(x) + \left(n + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 5\,n x - 4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(3\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	40	9						50
3	1	332	369	16					718
4	1	2672	9990	1952	25				14640
5	1	21396	233838	131360	7725	36			394356
6	1	171192	5151303	6917248	1117875	25704	49		13383372
7	1	1369564	110488455	322479280	115894675	7296804	76489	64	557605332

All 15 rows

Technical details

Node id: n-ee924b5c34b83af8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ffe5be3080cea858

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,       40,           9;
  3: 1,      332,         369,           16;
  4: 1,     2672,        9990,         1952,           25;
  5: 1,    21396,      233838,       131360,         7725,           36;
  6: 1,   171192,     5151303,      6917248,      1117875,        25704,          49;
  7: 1,  1369564,   110488455,    322479280,    115894675,      7296804,       76489,        64;
  8: 1, 10956544,  2341828188,  14077714720,  10052523250,   1368046512,    39968320,    211072,     81;
  9: 1, 87652388, 49375609740, 592251339584, 785392586750, 203710632552, 12764039792, 193759232, 552501, 100;

%F Formula

The row polynomials satisfy n*P(n,x) = ((8*n^2-8)*x+(9*n^3-2*n^2-7*n)*x^2)*P(n-2,x) + ((16*n+16)*x+(19*n^2+5*n-10)*x^2+(4*n^3-14*n^2-13*n)*x^3)*d/dx P(n-2,x) + ((26*n+26)*x^2+(43*n^2+31*n-2)*x^3-(4*n^2+3*n)*x^4)*d^2/dx^2 P(n-2,x) + ((6*n+6)*x^3+(12*n^2+9*n)*x^4)*d^3/dx^3 P(n-2,x) + (n+4*n^2*x)*P(n-1,x) + ((5*n-2)*x-(4*n^2+4*n)*x^2)*d/dx P(n-1,x) + 3*n*x^2*d^2/dx^2 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[(((8*n^2 - 8)*x + (9*n^3 - 2*n^2 - 7*n)*x^2)*P[n - 2, x] + ((16*n + 16)*x + (19*n^2 + 5*n - 10)*x^2 + (4*n^3 - 14*n^2 - 13*n)*x^3)*D[P[n - 2, x], x] + ((26*n + 26)*x^2 + (43*n^2 + 31*n - 2)*x^3 - (4*n^2 + 3*n)*x^4)*D[P[n - 2, x], {x, 2}] + ((6*n + 6)*x^3 + (12*n^2 + 9*n)*x^4)*D[P[n - 2, x], {x, 3}] + (n + 4*n^2*x)*P[n - 1, x] + ((5*n - 2)*x - (4*n^2 + 4*n)*x^2)*D[P[n - 1, x], x] + 3*n*x^2*D[P[n - 1, x], {x, 2}])/(n)]];
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] = (((8*n^2 - 8)*x + (9*n^3 - 2*n^2 - 7*n)*x^2)*v[i - 2] + ((16*n + 16)*x + (19*n^2 + 5*n - 10)*x^2 + (4*n^3 - 14*n^2 - 13*n)*x^3)*deriv(v[i - 2], x) + ((26*n + 26)*x^2 + (43*n^2 + 31*n - 2)*x^3 - (4*n^2 + 3*n)*x^4)*deriv(deriv(v[i - 2], x), x) + ((6*n + 6)*x^3 + (12*n^2 + 9*n)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (n + 4*n^2*x)*v[i - 1] + ((5*n - 2)*x - (4*n^2 + 4*n)*x^2)*deriv(v[i - 1], x) + 3*n*x^2*deriv(deriv(v[i - 1], x), x))/(n)); 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

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) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.