A232535 → (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

\(\left(27 + 24\,x + 3\,x^{2} - 36\,n + 29\,n x + 4\,n x^{2} - 114\,n^{2} x - 16\,n^{2} x^{2} + 88\,n^{3} x - 14\,n^{3} x^{2} + n^{3} x^{3} + 12\,n^{4} x^{2} + 4\,n^{4} x^{3} - 16\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n}(x) = \left(27 + 384\,x + 362\,x^{2} + 44\,x^{3} - x^{4} - 36\,n - 403\,n x - 535\,n x^{2} - 45\,n x^{3} - 5\,n x^{4} - 114\,n^{2} x - 76\,n^{2} x^{2} - 49\,n^{2} x^{3} - 5\,n^{2} x^{4} + 88\,n^{3} x + 442\,n^{3} x^{2} - 29\,n^{3} x^{3} + 31\,n^{3} x^{4} + 12\,n^{4} x^{2} - 84\,n^{4} x^{3} + 40\,n^{4} x^{4} - 16\,n^{5} x^{2} - 44\,n^{5} x^{3} + 12\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-72\,x - 34\,x^{2} + 78\,x^{3} + 30\,x^{4} - 2\,x^{5} + 168\,n x^{2} - 196\,n x^{3} + 42\,n x^{4} - 14\,n x^{5} + 176\,n^{2} x^{2} - 252\,n^{2} x^{3} + 114\,n^{2} x^{4} - 38\,n^{2} x^{5} - 96\,n^{3} x^{3} + 128\,n^{3} x^{4} - 32\,n^{3} x^{5} - 32\,n^{4} x^{3} + 40\,n^{4} x^{4} - 8\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

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

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	32	27													60
3	1	72	225	64												362
4	1	128	882	896	125											2032
5	1	200	2430	5376	2625	216										10848
6	1	288	5445	21120	22275	6336	343									55808
7	1	392	10647	64064	117975	72072	13377	512								279040
8	1	512	18900	163072	464750	494208	196196	25600	729							1363968
9	1	648	31212	365568	1491750	2450448	1689324	470016	45441	1000						6545408
10	1	800	48735	744192	4118250	9674496	10287550	4961280	1020357	76000	1331					30932992
11	1	968	72765	1404480	10126050	32232816	48968738	36382720	12950685	2048200	121121	1728				144310272
12	1	1152	104742	2493568	22713075	94140288	193659172	205396992	112527063	30764800	3857238	185472	2197			665845760
13	1	1352	146250	4209920	47279375	247464360	662518220	950912000	743160015	312455000	67654730	6888960	274625	2744		3042967552
14	1	1568	199017	6814080	92503125	596756160	2016783405	3765611520	3980569635	2386020000	793948155	139501440	11764935	395136	3375	13790871552

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Technical details

Node id: n-adf01e7484346696 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-0a1c85945a008331

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,  32,    27;
  3: 1,  72,   225,     64;
  4: 1, 128,   882,    896,     125;
  5: 1, 200,  2430,   5376,    2625,     216;
  6: 1, 288,  5445,  21120,   22275,    6336,     343;
  7: 1, 392, 10647,  64064,  117975,   72072,   13377,    512;
  8: 1, 512, 18900, 163072,  464750,  494208,  196196,  25600,   729;
  9: 1, 648, 31212, 365568, 1491750, 2450448, 1689324, 470016, 45441, 1000;

%F Formula

The row polynomials satisfy -(36*n-27-(88*n^3-114*n^2+29*n+24)*x+(16*n^5-12*n^4+14*n^3+16*n^2-4*n-3)*x^2-(4*n^5+4*n^4+n^3)*x^3)*P(n,x) = -(36*n-27-(88*n^3-114*n^2-403*n+384)*x+(16*n^5-12*n^4-442*n^3+76*n^2+535*n-362)*x^2+(44*n^5+84*n^4+29*n^3+49*n^2+45*n-44)*x^3-(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^4)*P(n-1,x) - (72*x-(176*n^2+168*n-34)*x^2+(32*n^4+96*n^3+252*n^2+196*n-78)*x^3-(40*n^4+128*n^3+114*n^2+42*n+30)*x^4+(8*n^4+32*n^3+38*n^2+14*n+2)*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[(-(36*n - 27 - (88*n^3 - 114*n^2 - 403*n + 384)*x + (16*n^5 - 12*n^4 - 442*n^3 + 76*n^2 + 535*n - 362)*x^2 + (44*n^5 + 84*n^4 + 29*n^3 + 49*n^2 + 45*n - 44)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*P[n - 1, x] - (72*x - (176*n^2 + 168*n - 34)*x^2 + (32*n^4 + 96*n^3 + 252*n^2 + 196*n - 78)*x^3 - (40*n^4 + 128*n^3 + 114*n^2 + 42*n + 30)*x^4 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*D[P[n - 1, x], x])/(-36*n + 27 + (88*n^3 - 114*n^2 + 29*n + 24)*x - (16*n^5 - 12*n^4 + 14*n^3 + 16*n^2 - 4*n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*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] = (-(36*n - 27 - (88*n^3 - 114*n^2 - 403*n + 384)*x + (16*n^5 - 12*n^4 - 442*n^3 + 76*n^2 + 535*n - 362)*x^2 + (44*n^5 + 84*n^4 + 29*n^3 + 49*n^2 + 45*n - 44)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*v[i - 1] - (72*x - (176*n^2 + 168*n - 34)*x^2 + (32*n^4 + 96*n^3 + 252*n^2 + 196*n - 78)*x^3 - (40*n^4 + 128*n^3 + 114*n^2 + 42*n + 30)*x^4 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*deriv(v[i - 1], x))/(-36*n + 27 + (88*n^3 - 114*n^2 + 29*n + 24)*x - (16*n^5 - 12*n^4 + 14*n^3 + 16*n^2 - 4*n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*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-ba353ec9512d926d
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-ba353ec9512d926d
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-ba353ec9512d926d
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-ba353ec9512d926d

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-ba353ec9512d926d

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a75bd3d9d36c6f35 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