A299989 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 9\,x + x^{2} - x^{3} + 15\,n x + 17\,n x^{2} + 5\,n x^{3} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n}(x) = \left(27 + 108\,x + 171\,x^{2} + 136\,x^{3} + 57\,x^{4} + 12\,x^{5} + x^{6} + 45\,n x + 135\,n x^{2} + 154\,n x^{3} + 83\,n x^{4} + 21\,n x^{5} + 2\,n x^{6} + 12\,n^{2} x^{2} + 28\,n^{2} x^{3} + 23\,n^{2} x^{4} + 8\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 4\,x\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	4															4
1	12	36	16													64
2	36	216	352	200	36											840
3	108	972	2736	3400	2052	588	64									9920
4	324	3888	15552	30000	31896	19600	6912	1296	100							109568
5	972	14580	75600	198000	303480	291256	179840	71280	17500	2420	144					1155072
6	2916	52488	334368	1107000	2221020	2914128	2600704	1605744	685500	198440	37152	4056	196			11763712
7	8748	183708	1388016	5556600	13791708	23004324	26975424	22800528	14049700	6311844	2043216	463736	69972	6300	256	116654080

All 15 rows

Technical details

Node id: n-3b0976fc57c747e3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-383c3e2053bee1bd

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 0,     4;
  1: 0,    12,      36,       16;
  2: 0,    36,     216,      352,       200,        36;
  3: 0,   108,     972,     2736,      3400,      2052,       588,         64;
  4: 0,   324,    3888,    15552,     30000,     31896,     19600,       6912,       1296,        100;
  5: 0,   972,   14580,    75600,    198000,    303480,    291256,     179840,      71280,      17500,       2420,        144;
  6: 0,  2916,   52488,   334368,   1107000,   2221020,   2914128,    2600704,    1605744,     685500,     198440,      37152,      4056,       196;
  7: 0,  8748,  183708,  1388016,   5556600,  13791708,  23004324,   26975424,   22800528,   14049700,    6311844,    2043216,    463736,     69972,     6300,      256;
  8: 0, 26244,  629856,  5505408,  25855200,  76504176, 154992096,  225630720,  242992224,  196893400,  120996128,   56407680,  19798688,   5142256,   957600,   120832,    9248,    324;
  9: 0, 78732, 2125764, 21100176, 113724000, 390825648, 932175216, 1622654208, 2127443616, 2143189800, 1678940824, 1028731104, 493193376, 184051056, 52844400, 11437056, 1803360, 195372, 12996, 400;

%F Formula

The row polynomials satisfy (9+(15*n+9)*x+(4*n^2+17*n+1)*x^2+(4*n^2+5*n-1)*x^3+n^2*x^4)*P(n,x) = (27+(45*n+108)*x+(12*n^2+135*n+171)*x^2+(28*n^2+154*n+136)*x^3+(23*n^2+83*n+57)*x^4+(8*n^2+21*n+12)*x^5+(n^2+2*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 4*x.

%t Mathematica

P[0, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*P[n - 1, x])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (45*n + 108)*x + (12*n^2 + 135*n + 171)*x^2 + (28*n^2 + 154*n + 136)*x^3 + (23*n^2 + 83*n + 57)*x^4 + (8*n^2 + 21*n + 12)*x^5 + (n^2 + 2*n + 1)*x^6)*v[i - 1])/(9 + (15*n + 9)*x + (4*n^2 + 17*n + 1)*x^2 + (4*n^2 + 5*n - 1)*x^3 + n^2*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - 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

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.