A391533 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 15\,x + 48\,x^{2} + 48\,x^{3} + 16\,x^{4} + 3\,n x + 20\,n x^{2} + 36\,n x^{3} + 16\,n x^{4} + n^{2} x^{2} + 4\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(1 + 19\,x + 87\,x^{2} + 157\,x^{3} + 124\,x^{4} + 36\,x^{5} + 3\,n x + 25\,n x^{2} + 66\,n x^{3} + 68\,n x^{4} + 24\,n x^{5} + n^{2} x^{2} + 5\,n^{2} x^{3} + 8\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 16\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	16	36															53
1	1	20	72	64														157
2	1	24	117	192	100													434
3	1	28	171	400	400	144												1144
4	1	32	234	704	1025	720	196											2912
5	1	36	306	1120	2125	2196	1176	256										7216
6	1	40	387	1664	3875	5256	4165	1792	324									17504
7	1	44	477	2352	6475	10836	11319	7232	2592	400								41728
8	1	48	576	3200	10150	20160	26068	22016	11745	3600	484							98048
9	1	52	684	4224	15150	34776	53508	56064	39609	18100	4840	576						227584
10	1	56	801	5440	21750	56592	100842	125952	110565	67000	26741	6336	676					522752
11	1	60	927	6864	30250	87912	177870	257664	269973	203500	107811	38160	8112	784				1189888
12	1	64	1062	8512	40975	131472	297528	489984	596079	536800	354046	166464	52897	10192	900			2686976
13	1	68	1206	10400	54275	190476	476476	878592	1216215	1272700	1003574	587808	248261	71540	12600	1024		6025216
14	1	72	1359	12544	70525	268632	735735	1500928	2328183	2774200	2543541	1782144	938119	359464	94725	15360	1156	13426688

Technical details

Node id: n-d4d0d20e032612ae · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8d0a5745cf4269c7

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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, 16,  36;
  1: 1, 20,  72,   64;
  2: 1, 24, 117,  192,   100;
  3: 1, 28, 171,  400,   400,   144;
  4: 1, 32, 234,  704,  1025,   720,   196;
  5: 1, 36, 306, 1120,  2125,  2196,  1176,   256;
  6: 1, 40, 387, 1664,  3875,  5256,  4165,  1792,   324;
  7: 1, 44, 477, 2352,  6475, 10836, 11319,  7232,  2592,   400;
  8: 1, 48, 576, 3200, 10150, 20160, 26068, 22016, 11745,  3600,  484;
  9: 1, 52, 684, 4224, 15150, 34776, 53508, 56064, 39609, 18100, 4840, 576;

%F Formula

The row polynomials satisfy (1+(3*n+15)*x+(n^2+20*n+48)*x^2+(4*n^2+36*n+48)*x^3+(4*n^2+16*n+16)*x^4)*P(n,x) = (1+(3*n+19)*x+(n^2+25*n+87)*x^2+(5*n^2+66*n+157)*x^3+(8*n^2+68*n+124)*x^4+(4*n^2+24*n+36)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 36*x^2.

%t Mathematica

P[0, x_] := 1 + 16*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 19)*x + (n^2 + 25*n + 87)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (8*n^2 + 68*n + 124)*x^4 + (4*n^2 + 24*n + 36)*x^5)*P[n - 1, x])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*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 + 16*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (3*n + 19)*x + (n^2 + 25*n + 87)*x^2 + (5*n^2 + 66*n + 157)*x^3 + (8*n^2 + 68*n + 124)*x^4 + (4*n^2 + 24*n + 36)*x^5)*v[i - 1])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*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 reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    no recorded property covers this requirement
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A159854 → basis A013609 → (θ+1)²

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.