A207606 → (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

\(\left(-24 + 516\,x - 426\,x^{2} + 24\,x^{3} - 8\,n + 564\,n x - 906\,n x^{2} + 88\,n x^{3} + 180\,n^{2} x - 696\,n^{2} x^{2} + 126\,n^{2} x^{3} + 20\,n^{3} x - 228\,n^{3} x^{2} + 88\,n^{3} x^{3} - 30\,n^{4} x^{2} + 30\,n^{4} x^{3} - 2\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n}(x) = \left(-32 + 592\,x + 680\,x^{2} - 736\,x^{3} + 36\,x^{4} - 8\,n + 668\,n x - 122\,n x^{2} - 1200\,n x^{3} + 114\,n x^{4} + 200\,n^{2} x - 582\,n^{2} x^{2} - 634\,n^{2} x^{3} + 136\,n^{2} x^{4} + 20\,n^{3} x - 234\,n^{3} x^{2} - 97\,n^{3} x^{3} + 76\,n^{3} x^{4} - 32\,n^{4} x^{2} + 12\,n^{4} x^{3} + 20\,n^{4} x^{4} - 2\,n^{5} x^{2} + 3\,n^{5} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-16\,x + 692\,x^{2} - 722\,x^{3} - 152\,x^{4} + 18\,x^{5} + 280\,n x^{2} - 1066\,n x^{3} - 92\,n x^{4} + 48\,n x^{5} + 40\,n^{2} x^{2} - 410\,n^{2} x^{3} + 71\,n^{2} x^{4} + 44\,n^{2} x^{5} - 56\,n^{3} x^{3} + 50\,n^{3} x^{4} + 16\,n^{3} x^{5} - 4\,n^{4} x^{3} + 7\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 4 + 28\,x + 18\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	4	28	18								50
1	5	64	99	32							200
2	6	120	324	240	50						740
3	7	200	819	1024	475	72					2597
4	8	308	1764	3264	2500	828	98				8770
5	9	448	3402	8640	9625	5184	1323	128			28759
6	10	624	6048	20064	30250	23400	9604	1984	162		92146
7	11	840	10098	42240	82225	85176	49735	16384	2835	200	289744

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d65430fb3bd1beab · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3e14a2e02426144d

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

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:  4,   28,    18;
  1:  5,   64,    99,     32;
  2:  6,  120,   324,    240,     50;
  3:  7,  200,   819,   1024,    475,     72;
  4:  8,  308,  1764,   3264,   2500,    828,     98;
  5:  9,  448,  3402,   8640,   9625,   5184,   1323,    128;
  6: 10,  624,  6048,  20064,  30250,  23400,   9604,   1984,    162;
  7: 11,  840, 10098,  42240,  82225,  85176,  49735,  16384,   2835,   200;
  8: 12, 1100, 16038,  82368, 200200, 265356, 205800,  95744,  26244,  3900,  242;
  9: 13, 1408, 24453, 151008, 446875, 733824, 723044, 443904, 170829, 40000, 5203, 288;

%F Formula

The row polynomials satisfy -(8*n+24-(20*n^3+180*n^2+564*n+516)*x+(2*n^5+30*n^4+228*n^3+696*n^2+906*n+426)*x^2-(4*n^5+30*n^4+88*n^3+126*n^2+88*n+24)*x^3)*P(n,x) = -(8*n+32-(20*n^3+200*n^2+668*n+592)*x+(2*n^5+32*n^4+234*n^3+582*n^2+122*n-680)*x^2-(3*n^5+12*n^4-97*n^3-634*n^2-1200*n-736)*x^3-(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^4)*P(n-1,x) - (16*x-(40*n^2+280*n+692)*x^2+(4*n^4+56*n^3+410*n^2+1066*n+722)*x^3-(7*n^4+50*n^3+71*n^2-92*n-152)*x^4-(2*n^4+16*n^3+44*n^2+48*n+18)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 28*x + 18*x^2.

%t Mathematica

P[0, x_] := 4 + 28*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*P[n - 1, x] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*D[P[n - 1, x], x])/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 28*x + 18*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*v[i - 1] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*deriv(v[i - 1], x))/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*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-8d9c8e34f3c4da6c
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-8d9c8e34f3c4da6c
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-8d9c8e34f3c4da6c
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-8d9c8e34f3c4da6c

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-8d9c8e34f3c4da6c

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the 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-62ac8b7225a515c7 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3222de37fcca30b3 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