A103327 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 - 3\,x + 18\,n + 5\,n x + 24\,n^{2} x - n^{2} x^{2} - 44\,n^{3} x - 2\,n^{3} x^{2} + 4\,n^{4} x^{2} + 8\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-9 - 111\,x - 35\,x^{2} - x^{3} + 18\,n + 221\,n x + 207\,n x^{2} - 6\,n x^{3} + 24\,n^{2} x - 161\,n^{2} x^{2} - 27\,n^{2} x^{3} - 44\,n^{3} x - 230\,n^{3} x^{2} + 42\,n^{3} x^{3} + 4\,n^{4} x^{2} + 76\,n^{4} x^{3} + 8\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(36\,x - 22\,x^{2} - 12\,x^{3} - 2\,x^{4} - 128\,n x^{2} + 144\,n x^{3} - 16\,n x^{4} - 88\,n^{2} x^{2} + 156\,n^{2} x^{3} - 68\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4} + 16\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

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	12														13
2	1	40	45													86
3	1	84	315	112												512
4	1	144	1134	1344	225											2848
5	1	220	2970	7392	4125	396										15104
6	1	312	6435	27456	32175	10296	637									77312
7	1	420	12285	80080	160875	108108	22295	960								385024
8	1	544	21420	198016	607750	700128	303212	43520	1377							1875968
9	1	684	34884	434112	1889550	3325608	2469012	744192	78489	1900						8978432
10	1	840	53865	868224	5087250	12697776	14402570	7441920	1648269	133000	2541					42336256
11	1	1012	79695	1615152	12257850	41186376	66251822	52300160	19857717	3364900	214291	3312				197132288
12	1	1200	113850	2833600	27039375	117675360	254814700	285273600	165480975	48070000	6428730	331200	4225			908066816
13	1	1404	157950	4736160	55501875	303706260	851809140	1283731200	1056069495	468682500	107451630	11625120	494325	5292		4143972352
14	1	1624	213759	7600320	107303625	721080360	2542900815	4963760640	5496977115	3459729000	1211815605	224752320	20069595	716184	6525	18756927488

Technical details

Node id: n-7aa808e384d6bebf · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d8c33c6b14bc1f96

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,  12;
  2: 1,  40,    45;
  3: 1,  84,   315,    112;
  4: 1, 144,  1134,   1344,     225;
  5: 1, 220,  2970,   7392,    4125,     396;
  6: 1, 312,  6435,  27456,   32175,   10296,     637;
  7: 1, 420, 12285,  80080,  160875,  108108,   22295,    960;
  8: 1, 544, 21420, 198016,  607750,  700128,  303212,  43520,  1377;
  9: 1, 684, 34884, 434112, 1889550, 3325608, 2469012, 744192, 78489, 1900;

%F Formula

The row polynomials satisfy (18*n-9-(44*n^3-24*n^2-5*n+3)*x+(8*n^5+4*n^4-2*n^3-n^2)*x^2)*P(n,x) = (18*n-9-(44*n^3-24*n^2-221*n+111)*x+(8*n^5+4*n^4-230*n^3-161*n^2+207*n-35)*x^2+(24*n^5+76*n^4+42*n^3-27*n^2-6*n-1)*x^3)*P(n-1,x) + (36*x-(88*n^2+128*n+22)*x^2+(16*n^4+64*n^3+156*n^2+144*n-12)*x^3-(16*n^4+64*n^3+68*n^2+16*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n - 9 - (44*n^3 - 24*n^2 - 221*n + 111)*x + (8*n^5 + 4*n^4 - 230*n^3 - 161*n^2 + 207*n - 35)*x^2 + (24*n^5 + 76*n^4 + 42*n^3 - 27*n^2 - 6*n - 1)*x^3)*P[n - 1, x] + (36*x - (88*n^2 + 128*n + 22)*x^2 + (16*n^4 + 64*n^3 + 156*n^2 + 144*n - 12)*x^3 - (16*n^4 + 64*n^3 + 68*n^2 + 16*n + 2)*x^4)*D[P[n - 1, x], x])/(18*n - 9 - (44*n^3 - 24*n^2 - 5*n + 3)*x + (8*n^5 + 4*n^4 - 2*n^3 - n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((18*n - 9 - (44*n^3 - 24*n^2 - 221*n + 111)*x + (8*n^5 + 4*n^4 - 230*n^3 - 161*n^2 + 207*n - 35)*x^2 + (24*n^5 + 76*n^4 + 42*n^3 - 27*n^2 - 6*n - 1)*x^3)*v[i - 1] + (36*x - (88*n^2 + 128*n + 22)*x^2 + (16*n^4 + 64*n^3 + 156*n^2 + 144*n - 12)*x^3 - (16*n^4 + 64*n^3 + 68*n^2 + 16*n + 2)*x^4)*deriv(v[i - 1], x))/(18*n - 9 - (44*n^3 - 24*n^2 - 5*n + 3)*x + (8*n^5 + 4*n^4 - 2*n^3 - n^2)*x^2)); 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 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A091042 → (θ+1)²

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023