A121574 → (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

\(\left(-32\,n + 96\,n x + 16\,n x^{3} - 32\,n^{2} + 32\,n^{2} x + 120\,n^{2} x^{2} + 16\,n^{2} x^{3} - 4\,n^{2} x^{4} - 72\,n^{3} x + 42\,n^{3} x^{2} + 39\,n^{3} x^{3} - 3\,n^{3} x^{4} - 18\,n^{4} x^{2} + 9\,n^{4} x^{3}\right)\,P_{n}(x) = \left(256\,x + 208\,x^{2} + 144\,x^{3} + 40\,x^{4} - 8\,x^{5} - 64\,n + 64\,n x + 616\,n x^{2} + 392\,n x^{3} + 106\,n x^{4} - 14\,n x^{5} - 64\,n^{2} - 320\,n^{2} x + 284\,n^{2} x^{3} + 121\,n^{2} x^{4} - 6\,n^{2} x^{5} - 144\,n^{3} x - 276\,n^{3} x^{2} - 48\,n^{3} x^{3} + 39\,n^{3} x^{4} - 36\,n^{4} x^{2} - 36\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(64\,x + 192\,x^{2} + 176\,x^{3} + 64\,x^{4} - 24\,x^{5} - 4\,x^{6} + 64\,n x + 384\,n x^{2} + 632\,n x^{3} + 244\,n x^{4} - 30\,n x^{5} - 7\,n x^{6} + 144\,n^{2} x^{2} + 420\,n^{2} x^{3} + 288\,n^{2} x^{4} + 9\,n^{2} x^{5} - 3\,n^{2} x^{6} + 36\,n^{3} x^{3} + 72\,n^{3} x^{4} + 9\,n^{3} x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	4	20	9						33
3	8	64	72	16					160
4	16	176	333	176	25				726
5	32	448	1206	1072	350	36			3144
6	64	1088	3816	4880	2650	612	49		13159
7	128	2560	11088	18688	14600	5544	980	64	53652

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-4b7add9985ac0be4 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e6fcc40b70107e9e

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:   2,     4;
  2:   4,    20,     9;
  3:   8,    64,    72,     16;
  4:  16,   176,   333,    176,     25;
  5:  32,   448,  1206,   1072,    350,     36;
  6:  64,  1088,  3816,   4880,   2650,    612,    49;
  7: 128,  2560, 11088,  18688,  14600,   5544,   980,    64;
  8: 256,  5888, 30384,  63872,  66025,  35928, 10339,  1472,   81;
  9: 512, 13312, 79776, 201472, 261050, 187956, 77126, 17728, 2106, 100;

%F Formula

The row polynomials satisfy -(32*n^2+32*n+(72*n^3-32*n^2-96*n)*x+(18*n^4-42*n^3-120*n^2)*x^2-(9*n^4+39*n^3+16*n^2+16*n)*x^3+(3*n^3+4*n^2)*x^4)*P(n,x) = -(64*n^2+64*n+(144*n^3+320*n^2-64*n-256)*x+(36*n^4+276*n^3-616*n-208)*x^2+(36*n^4+48*n^3-284*n^2-392*n-144)*x^3-(39*n^3+121*n^2+106*n+40)*x^4+(6*n^2+14*n+8)*x^5)*P(n-1,x) + ((64*n+64)*x+(144*n^2+384*n+192)*x^2+(36*n^3+420*n^2+632*n+176)*x^3+(72*n^3+288*n^2+244*n+64)*x^4+(9*n^3+9*n^2-30*n-24)*x^5-(3*n^2+7*n+4)*x^6)*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[(-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*P[n - 1, x] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*D[P[n - 1, x], x])/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*v[i - 1] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*deriv(v[i - 1], x))/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*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)

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-01576e9f8151fd70
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-01576e9f8151fd70
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-01576e9f8151fd70
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-01576e9f8151fd70

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-01576e9f8151fd70

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-81aa6611c032ec70 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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
[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.