A116414 → (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

\(\left(128\,n + 32\,n x + 32\,n x^{2} - 80\,n^{3} x^{2} - 40\,n^{3} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(256 + 208\,x + 144\,x^{2} + 40\,x^{3} - 8\,x^{4} + 256\,n + 320\,n x + 128\,n x^{2} - 28\,n x^{3} - 44\,n x^{4} + 112\,n^{2} x - 176\,n^{2} x^{2} - 208\,n^{2} x^{3} - 72\,n^{2} x^{4} + n^{2} x^{5} - 160\,n^{3} x^{2} - 148\,n^{3} x^{3} - 36\,n^{3} x^{4} + 3\,n^{3} x^{5} - 8\,n^{4} x^{3} + 4\,n^{4} x^{4} + 3\,n^{4} x^{5} + 4\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(64 + 192\,x + 176\,x^{2} + 64\,x^{3} - 24\,x^{4} - 4\,x^{5} - 112\,n x^{2} - 304\,n x^{3} - 188\,n x^{4} - 20\,n x^{5} - 112\,n^{2} x^{2} - 304\,n^{2} x^{3} - 184\,n^{2} x^{4} - 12\,n^{2} x^{5} + n^{2} x^{6} + 8\,n^{3} x^{4} + 16\,n^{3} x^{5} + 2\,n^{3} x^{6} + 4\,n^{4} x^{4} + 8\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	4							8
2	13	32	9						54
3	40	168	108	16					332
4	121	736	783	256	25				1921
5	364	2924	4464	2368	500	36			10656
6	1093	10944	22086	16640	5625	864	49		57301
7	3280	39376	99576	98720	47000	11448	1372	64	300836

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8bb8b4f25546250f · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3f56d13b5db11e63

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:     4,      4;
  2:    13,     32,       9;
  3:    40,    168,     108,      16;
  4:   121,    736,     783,     256,      25;
  5:   364,   2924,    4464,    2368,     500,     36;
  6:  1093,  10944,   22086,   16640,    5625,    864,     49;
  7:  3280,  39376,   99576,   98720,   47000,  11448,   1372,    64;
  8:  9841, 137792,  420642,  521984,  325375, 110880,  20923,  2048,   81;
  9: 29524, 472404, 1693872, 2539584, 1976100, 877716, 230496, 35328, 2916, 100;

%F Formula

The row polynomials satisfy (128*n+32*n*x-(80*n^3-32*n)*x^2-40*n^3*x^3+2*n^5*x^4)*P(n,x) = (256*n+256+(112*n^2+320*n+208)*x-(160*n^3+176*n^2-128*n-144)*x^2-(8*n^4+148*n^3+208*n^2+28*n-40)*x^3+(4*n^5+4*n^4-36*n^3-72*n^2-44*n-8)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (64+192*x-(112*n^2+112*n-176)*x^2-(304*n^2+304*n-64)*x^3+(4*n^4+8*n^3-184*n^2-188*n-24)*x^4+(8*n^4+16*n^3-12*n^2-20*n-4)*x^5+(n^4+2*n^3+n^2)*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[((256*n + 256 + (112*n^2 + 320*n + 208)*x - (160*n^3 + 176*n^2 - 128*n - 144)*x^2 - (8*n^4 + 148*n^3 + 208*n^2 + 28*n - 40)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (64 + 192*x - (112*n^2 + 112*n - 176)*x^2 - (304*n^2 + 304*n - 64)*x^3 + (4*n^4 + 8*n^3 - 184*n^2 - 188*n - 24)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(128*n + 32*n*x - (80*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*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] = ((256*n + 256 + (112*n^2 + 320*n + 208)*x - (160*n^3 + 176*n^2 - 128*n - 144)*x^2 - (8*n^4 + 148*n^3 + 208*n^2 + 28*n - 40)*x^3 + (4*n^5 + 4*n^4 - 36*n^3 - 72*n^2 - 44*n - 8)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (64 + 192*x - (112*n^2 + 112*n - 176)*x^2 - (304*n^2 + 304*n - 64)*x^3 + (4*n^4 + 8*n^3 - 184*n^2 - 188*n - 24)*x^4 + (8*n^4 + 16*n^3 - 12*n^2 - 20*n - 4)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(128*n + 32*n*x - (80*n^3 - 32*n)*x^2 - 40*n^3*x^3 + 2*n^5*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-ec090eec720f3720
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-ec090eec720f3720
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-ec090eec720f3720
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-ec090eec720f3720

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-ec090eec720f3720

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-894010de7c322dc8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-7bedda0241d23786 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