A111125 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-24 + 6\,x + 3\,x^{2} + 25\,n - 4\,n x - 2\,n x^{2} - 9\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(-6 + 4\,x^{2} + x^{3} + 13\,n - n x - 5\,n x^{2} - n x^{3} - 3\,n^{2} - 2\,n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 13\,x^{2} - 8\,x^{3} - x^{4} - 8\,n x - 2\,n x^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	10	7	1													18
4	21	28	10	1												60
5	39	84	54	13	1											191
6	66	209	209	88	16	1										589
7	104	455	650	416	130	19	1									1775
8	155	896	1728	1550	725	180	22	1								5257
9	221	1632	4080	4862	3145	1156	238	25	1							15360
10	304	2793	8778	13376	11362	5719	1729	304	28	1						44394
11	406	4543	17523	33176	35672	23373	9604	2464	378	31	1					127171
12	529	7084	32890	75647	100165	82432	43792	15180	3381	460	34	1				361595
13	675	10660	58630	160875	256750	258400	171360	76380	22875	4500	550	37	1			1021693
14	846	15561	100035	322608	609960	735318	593028	328320	125874	33165	5841	648	40	1		2871245
15	1044	22127	164372	615264	1358708	1929602	1854666	1243056	589512	198099	46574	7424	754	43	1	8031246

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Technical details

Node id: n-5ad11fc70a3588cf · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5aabb12a2c39c57b

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 -1)
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 ?

A377857: 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   4,    1;
   3:  10,    7,    1;
   4:  21,   28,   10,     1;
   5:  39,   84,   54,    13,     1;
   6:  66,  209,  209,    88,    16,    1;
   7: 104,  455,  650,   416,   130,   19,    1;
   8: 155,  896, 1728,  1550,   725,  180,   22,   1;
   9: 221, 1632, 4080,  4862,  3145, 1156,  238,  25,  1;
  10: 304, 2793, 8778, 13376, 11362, 5719, 1729, 304, 28, 1;

%F Formula

The row polynomials satisfy (2*n^3-9*n^2+25*n-24-(4*n-6)*x-(2*n-3)*x^2)*P(n,x) = (2*n^3-3*n^2+13*n-6+(n^3-2*n^2-n)*x-(5*n-4)*x^2-(n-1)*x^3)*P(n-1,x) + ((4*n^2-8*n+12)*x+(n^2-2*n-13)*x^2-8*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 3*n^2 + 13*n - 6 + (n^3 - 2*n^2 - n)*x - (5*n - 4)*x^2 - (n - 1)*x^3)*P[n - 1, x] + ((4*n^2 - 8*n + 12)*x + (n^2 - 2*n - 13)*x^2 - 8*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 25*n - 24 - (4*n - 6)*x - (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^3 - 3*n^2 + 13*n - 6 + (n^3 - 2*n^2 - n)*x - (5*n - 4)*x^2 - (n - 1)*x^3)*v[i - 1] + ((4*n^2 - 8*n + 12)*x + (n^2 - 2*n - 13)*x^2 - 8*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 25*n - 24 - (4*n - 6)*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-23729e33b528c150
Coefficients of one sign. Every nonzero row, n >= 1: 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-23729e33b528c150
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-23729e33b528c150
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-23729e33b528c150

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 derivative_polynomial, shift 0 e-23729e33b528c150

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-0bb8ed02ac535b32 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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