A108838 → 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

\(\left(3 + 4\,n + 3\,n x + n^{2} + 7\,n^{2} x + 5\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(6 + 5\,n + 16\,n x + 3\,n x^{2} + n^{2} + 17\,n^{2} x + 14\,n^{2} x^{2} + 6\,n^{3} x + 12\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{3} + 2\,n x + 8\,n x^{2} - 10\,n x^{3} + 8\,n^{2} x^{2} - 8\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	4														7
2	4	16	6													26
3	5	40	45	8												98
4	6	80	180	96	10											372
5	7	140	525	560	175	12										1419
6	8	224	1260	2240	1400	288	14									5434
7	9	336	2646	7056	7350	3024	441	16								20878
8	10	480	5040	18816	29400	20160	5880	640	18							80444
9	11	660	8910	44352	97020	99792	48510	10560	891	20						310726
10	12	880	14850	95040	277200	399168	291060	105600	17820	1200	22					1202852
11	13	1144	23595	188760	707850	1359072	1387386	755040	212355	28600	1573	24				4665412
12	14	1456	36036	352352	1651650	4077216	5549544	4228224	1783782	400400	44044	2016	26			18126760
13	15	1820	53235	624624	3578575	11042460	19324305	19631040	11594583	3903900	715715	65520	2535	28		70538355
14	16	2240	76440	1059968	7287280	27483456	60120060	78524160	61837776	29149120	8016008	1223040	94640	3136	30	274877370

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Technical details

Node id: n-a5bcbfbd7519fc4f · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b29d29aac1e7588a

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 ?

A126223: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  2;
  1:  3,   4;
  2:  4,  16,    6;
  3:  5,  40,   45,     8;
  4:  6,  80,  180,    96,    10;
  5:  7, 140,  525,   560,   175,    12;
  6:  8, 224, 1260,  2240,  1400,   288,    14;
  7:  9, 336, 2646,  7056,  7350,  3024,   441,    16;
  8: 10, 480, 5040, 18816, 29400, 20160,  5880,   640,  18;
  9: 11, 660, 8910, 44352, 97020, 99792, 48510, 10560, 891, 20;

%F Formula

The row polynomials satisfy (n^2+4*n+3+(n^4+5*n^3+7*n^2+3*n)*x)*P(n,x) = (n^2+5*n+6+(n^4+6*n^3+17*n^2+16*n)*x+(3*n^4+12*n^3+14*n^2+3*n)*x^2)*P(n-1,x) + ((2*n+3)*x+(2*n^3+8*n^2+8*n)*x^2-(2*n^3+8*n^2+10*n+3)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (n^4 + 6*n^3 + 17*n^2 + 16*n)*x + (3*n^4 + 12*n^3 + 14*n^2 + 3*n)*x^2)*P[n - 1, x] + ((2*n + 3)*x + (2*n^3 + 8*n^2 + 8*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 3)*x^3)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (n^4 + 5*n^3 + 7*n^2 + 3*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (n^4 + 6*n^3 + 17*n^2 + 16*n)*x + (3*n^4 + 12*n^3 + 14*n^2 + 3*n)*x^2)*v[i - 1] + ((2*n + 3)*x + (2*n^3 + 8*n^2 + 8*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 3)*x^3)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (n^4 + 5*n^3 + 7*n^2 + 3*n)*x)); 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-12dbe0b95f16e416
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-12dbe0b95f16e416
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-12dbe0b95f16e416
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-12dbe0b95f16e416

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-12dbe0b95f16e416

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of Dyck paths counted by number of long interior inclines.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a4549745e8cd4154 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