A125662 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20\,n - 6\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-30 - 24\,x - 6\,x^{2} - 30\,n - 29\,n x - 6\,n x^{2} + n x^{3} - 5\,n^{2} x + 3\,n^{2} x^{2} + 2\,n^{2} x^{3} + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 45\,x - 23\,x^{2} - 3\,x^{3} + 5\,n x^{2} + 6\,n x^{3} + n x^{4} + 5\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	8	12	3													23
3	21	50	27	4												102
4	55	180	153	48	5											441
5	144	600	702	344	75	6										1871
6	377	1908	2853	1920	650	108	7									7823
7	987	5878	10719	9220	4275	1098	147	8								32332
8	2584	17700	38121	40032	23700	8316	1715	192	9							132369
9	6765	52390	130194	161704	116865	52290	14700	2528	243	10						537689
10	17711	153000	431046	618672	529025	288792	103390	24192	3564	300	11					2169703
11	46368	442032	1392444	2269040	2243550	1448916	632394	188208	37665	4850	363	12				8705842
12	121393	1265832	4409334	8045040	9037925	6750216	3484194	1263168	321165	56100	6413	432	13			34761225
13	317811	3598250	13734306	27745192	34926525	29647242	17695440	7583232	2346894	520300	80586	8280	507	14		138204579
14	832040	10164540	42190959	93502944	130423140	124103340	84190575	41709888	15250653	4112460	807675	112320	10478	588	15	547411615

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Technical details

Node id: n-d01ecaaeb3718f4e · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-754d74333aa59564

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:    3,     2;
  2:    8,    12,      3;
  3:   21,    50,     27,      4;
  4:   55,   180,    153,     48,      5;
  5:  144,   600,    702,    344,     75,     6;
  6:  377,  1908,   2853,   1920,    650,   108,     7;
  7:  987,  5878,  10719,   9220,   4275,  1098,   147,    8;
  8: 2584, 17700,  38121,  40032,  23700,  8316,  1715,  192,   9;
  9: 6765, 52390, 130194, 161704, 116865, 52290, 14700, 2528, 243, 10;

%F Formula

The row polynomials satisfy -(20*n+6*n*x-2*n^3*x^2)*P(n,x) = -(30*n+30+(5*n^2+29*n+24)*x-(3*n^3+3*n^2-6*n-6)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (25+45*x-(5*n^2+5*n-23)*x^2-(6*n^2+6*n-3)*x^3-(n^2+n)*x^4)*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[(-(30*n + 30 + (5*n^2 + 29*n + 24)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (25 + 45*x - (5*n^2 + 5*n - 23)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-20*n - 6*n*x + 2*n^3*x^2)]];
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] = (-(30*n + 30 + (5*n^2 + 29*n + 24)*x - (3*n^3 + 3*n^2 - 6*n - 6)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (25 + 45*x - (5*n^2 + 5*n - 23)*x^2 - (6*n^2 + 6*n - 3)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-20*n - 6*n*x + 2*n^3*x^2)); 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-997ef18dd206f0e8
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-997ef18dd206f0e8
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-997ef18dd206f0e8
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-997ef18dd206f0e8

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-997ef18dd206f0e8

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-af8f298ca8f41adf root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-c7df247a48c8552e 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