A111577 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - 4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-9\,x^{2} + 12\,x^{3}\right)\,P_{n-2}''(x) + \left(-1 + 4\,x\right)\,P_{n-1}(x) + \left(6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	20	9													30
4	1	84	108	16												209
5	1	340	945	352	25											1663
6	1	1364	7380	5200	875	36										14856
7	1	5460	54729	65120	19500	1836	49									146695
8	1	21844	395388	748496	355250	57456	3430	64								1581929
9	1	87380	2816865	8187872	5787775	1430856	143374	5888	81							18460092
10	1	349524	19914660	86886480	88034625	31228092	4671660	316800	9477	100						231411419
11	1	1398100	140189049	904268640	1280210250	626419332	131266443	13071360	637875	14500	121					3097475671
12	1	5592404	984469068	9291911376	18055653000	11866212072	3346688730	459019968	32490315	1193500	21296	144				44043251874
13	1	22369620	6903866385	94669280992	249242100525	215859533472	79738318968	14469624576	1393205022	73529500	2104311	30240	169			662373963781
14	1	89478484	48377396340	958966350160	3388068058375	3812661160308	1808836869840	422479749120	53143244154	3778832200	154204336	3532464	41743	196		10496558917721
15	1	357913940	338843100969	9675667761760	45543269681000	65881396568988	39557355995157	11657116922880	1863282036330	171416245000	9352721378	303619680	5690230	56252	225	174698368313790

Technical details

Node id: n-448c6b1edeb577e3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-66c5f01cd4a0fcbb

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 -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;
   2: 1,      4;
   3: 1,     20,        9;
   4: 1,     84,      108,       16;
   5: 1,    340,      945,      352,       25;
   6: 1,   1364,     7380,     5200,      875,       36;
   7: 1,   5460,    54729,    65120,    19500,     1836,      49;
   8: 1,  21844,   395388,   748496,   355250,    57456,    3430,     64;
   9: 1,  87380,  2816865,  8187872,  5787775,  1430856,  143374,   5888,   81;
  10: 1, 349524, 19914660, 86886480, 88034625, 31228092, 4671660, 316800, 9477, 100;

%F Formula

The row polynomials satisfy P(n,x) = (2-4*x+9*x^2)*P(n-2,x) - (6*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (9*x^2-12*x^3)*d^2/dx^2 P(n-2,x) - (1-4*x)*P(n-1,x) + (6*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 - 4*x + 9*x^2)*P[n - 2, x] - (6*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (9*x^2 - 12*x^3)*D[P[n - 2, x], {x, 2}] - (1 - 4*x)*P[n - 1, x] + (6*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (2 - 4*x + 9*x^2)*v[i - 2] - (6*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) - (1 - 4*x)*v[i - 1] + (6*x - 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, 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 Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.