A207607 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-36 - 438\,x + 84\,x^{2} - 72\,n + 346\,n x - 356\,n x^{2} + 66\,n^{2} x + 395\,n^{2} x^{2} - 16\,n^{2} x^{3} + 44\,n^{3} x - 116\,n^{3} x^{2} + 32\,n^{3} x^{3} - 5\,n^{4} x^{2} - 20\,n^{4} x^{3} - 2\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n}(x) = \left(-36 - 1158\,x - 596\,x^{2} - 112\,x^{3} - 8\,x^{4} - 72\,n + 202\,n x + 164\,n x^{2} - 2\,n x^{3} + 66\,n^{2} x + 661\,n^{2} x^{2} + 204\,n^{2} x^{3} + 14\,n^{2} x^{4} + 44\,n^{3} x - 78\,n^{3} x^{2} - 64\,n^{3} x^{3} - 2\,n^{3} x^{4} - 5\,n^{4} x^{2} - 29\,n^{4} x^{3} - 6\,n^{4} x^{4} - 2\,n^{5} x^{2} + 3\,n^{5} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-144\,x + 988\,x^{2} + 568\,x^{3} + 110\,x^{4} + 8\,x^{5} + 176\,n x^{2} + 348\,n x^{3} + 108\,n x^{4} + 8\,n x^{5} + 88\,n^{2} x^{2} - 250\,n^{2} x^{3} - 92\,n^{2} x^{4} - 6\,n^{2} x^{5} - 16\,n^{3} x^{3} - 20\,n^{3} x^{4} - 4\,n^{3} x^{5} - 4\,n^{4} x^{3} + 7\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

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

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	8														9
2	1	20	18													39
3	1	36	81	32												150
4	1	56	225	208	50											540
5	1	80	495	784	425	72										1857
6	1	108	945	2240	2025	756	98									6173
7	1	140	1638	5376	7125	4356	1225	128								19989
8	1	176	2646	11424	20625	18216	8281	1856	162							63387
9	1	216	4050	22176	51975	61776	40131	14400	2673	200						197598
10	1	260	5940	40128	117975	180180	156065	79360	23409	3700	242					607260
11	1	308	8415	68640	246675	468468	517244	348160	144585	36100	4961	288				1843845
12	1	360	11583	112112	482625	1111968	1516060	1292544	706401	247000	53361	6480	338			5540833
13	1	416	15561	176176	893750	2450448	4028388	4217088	2904093	1330000	400631	76176	8281	392		16501401
14	1	476	20475	267904	1580150	5075928	9876048	12403200	10439037	5998300	2357201	622656	105625	10388	450	48757839

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Technical details

Node id: n-ecbdd4aafecd6d87 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-9d3366654f7fca27

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,   8;
  2: 1,  20,   18;
  3: 1,  36,   81,    32;
  4: 1,  56,  225,   208,    50;
  5: 1,  80,  495,   784,   425,    72;
  6: 1, 108,  945,  2240,  2025,   756,    98;
  7: 1, 140, 1638,  5376,  7125,  4356,  1225,   128;
  8: 1, 176, 2646, 11424, 20625, 18216,  8281,  1856,  162;
  9: 1, 216, 4050, 22176, 51975, 61776, 40131, 14400, 2673, 200;

%F Formula

The row polynomials satisfy -(72*n+36-(44*n^3+66*n^2+346*n-438)*x+(2*n^5+5*n^4+116*n^3-395*n^2+356*n-84)*x^2-(4*n^5-20*n^4+32*n^3-16*n^2)*x^3)*P(n,x) = -(72*n+36-(44*n^3+66*n^2+202*n-1158)*x+(2*n^5+5*n^4+78*n^3-661*n^2-164*n+596)*x^2-(3*n^5-29*n^4-64*n^3+204*n^2-2*n-112)*x^3-(2*n^5-6*n^4-2*n^3+14*n^2-8)*x^4)*P(n-1,x) - (144*x-(88*n^2+176*n+988)*x^2+(4*n^4+16*n^3+250*n^2-348*n-568)*x^3-(7*n^4-20*n^3-92*n^2+108*n+110)*x^4-(2*n^4-4*n^3-6*n^2+8*n+8)*x^5)*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[(-(72*n + 36 - (44*n^3 + 66*n^2 + 202*n - 1158)*x + (2*n^5 + 5*n^4 + 78*n^3 - 661*n^2 - 164*n + 596)*x^2 - (3*n^5 - 29*n^4 - 64*n^3 + 204*n^2 - 2*n - 112)*x^3 - (2*n^5 - 6*n^4 - 2*n^3 + 14*n^2 - 8)*x^4)*P[n - 1, x] - (144*x - (88*n^2 + 176*n + 988)*x^2 + (4*n^4 + 16*n^3 + 250*n^2 - 348*n - 568)*x^3 - (7*n^4 - 20*n^3 - 92*n^2 + 108*n + 110)*x^4 - (2*n^4 - 4*n^3 - 6*n^2 + 8*n + 8)*x^5)*D[P[n - 1, x], x])/(-72*n - 36 + (44*n^3 + 66*n^2 + 346*n - 438)*x - (2*n^5 + 5*n^4 + 116*n^3 - 395*n^2 + 356*n - 84)*x^2 + (4*n^5 - 20*n^4 + 32*n^3 - 16*n^2)*x^3)]];
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] = (-(72*n + 36 - (44*n^3 + 66*n^2 + 202*n - 1158)*x + (2*n^5 + 5*n^4 + 78*n^3 - 661*n^2 - 164*n + 596)*x^2 - (3*n^5 - 29*n^4 - 64*n^3 + 204*n^2 - 2*n - 112)*x^3 - (2*n^5 - 6*n^4 - 2*n^3 + 14*n^2 - 8)*x^4)*v[i - 1] - (144*x - (88*n^2 + 176*n + 988)*x^2 + (4*n^4 + 16*n^3 + 250*n^2 - 348*n - 568)*x^3 - (7*n^4 - 20*n^3 - 92*n^2 + 108*n + 110)*x^4 - (2*n^4 - 4*n^3 - 6*n^2 + 8*n + 8)*x^5)*deriv(v[i - 1], x))/(-72*n - 36 + (44*n^3 + 66*n^2 + 346*n - 438)*x - (2*n^5 + 5*n^4 + 116*n^3 - 395*n^2 + 356*n - 84)*x^2 + (4*n^5 - 20*n^4 + 32*n^3 - 16*n^2)*x^3)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-6f5daa3d60a537b3
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-6f5daa3d60a537b3
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-6f5daa3d60a537b3
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-6f5daa3d60a537b3

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-6f5daa3d60a537b3

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8f881604bf5a9d75 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3cffc405672d7710 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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.
[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