A144696 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8\,n x + 16\,n x^{2} - 4\,n^{2} x + 25\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 17\,x^{2} - 23\,x^{3} - 2\,n x + 16\,n x^{2} - 29\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 9\,x^{3} - 7\,x^{4} - n x^{2} + 12\,n x^{3} - 11\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,x + n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} + 2\,n x - 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	8														9
2	1	28	36													65
3	1	72	297	128												498
4	1	164	1539	2096	400											4200
5	1	352	6462	19328	11825	1152										39120
6	1	732	24138	134752	179925	57996	3136									400680
7	1	1496	83943	796480	1952375	1384344	258769	8192								4485600
8	1	3028	278757	4230544	17228875	22363164	9348367	1079488	20736							54552960
9	1	6096	897588	20886720	132415950	283036464	217632324	57476352	4285305	51200						716688000
10	1	12236	2829924	97908288	923163750	3032971560	3834890724	1881083136	329541777	16383500	123904					10118908800
11	1	24520	8792613	441942912	5992654050	28832675760	55741714098	45101686272	14869605213	1791201800	60835049	294912				152841427200
12	1	49092	27039879	1939716080	36868628250	250660851048	704147801406	870452019072	476317377045	109699704500	9338602251	220735152	692224			2459673216000
13	1	98240	82555578	8335715072	217682011375	2034873353088	7999630035180	14321242159104	11998517249871	4625273912000	766407836986	47103655680	786170593	1605632		42019936358400
14	1	196540	250761294	35250811424	1244704714475	15657323178708	83694297552180	208606404625152	252989232109767	149943628933700	42009974807966	5125687195680	231345712429	2757794284	3686400	759540862080000

Technical details

Node id: n-56035fcd2a59cb1a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-292cac4344de9c25

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,    8;
  2: 1,   28,     36;
  3: 1,   72,    297,      128;
  4: 1,  164,   1539,     2096,       400;
  5: 1,  352,   6462,    19328,     11825,      1152;
  6: 1,  732,  24138,   134752,    179925,     57996,      3136;
  7: 1, 1496,  83943,   796480,   1952375,   1384344,    258769,     8192;
  8: 1, 3028, 278757,  4230544,  17228875,  22363164,   9348367,  1079488,   20736;
  9: 1, 6096, 897588, 20886720, 132415950, 283036464, 217632324, 57476352, 4285305, 51200;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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 2-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.