A137320 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + 8\,x - 4\,x^{2} + 4\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-3 + 4\,x + 2\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	2	10	4													16
3	4	28	36	8												76
4	12	100	188	112	16											428
5	48	448	1040	920	320	32										2808
6	240	2432	6496	7240	3760	864	64									21096
7	1440	15552	46032	60032	39760	13664	2240	128								178848
8	10080	114624	367440	538272	421904	186368	45696	5632	256							1690272
9	80640	957312	3273312	5251168	4668384	2460192	779520	143616	13824	512						17628480
10	725760	8938368	32251104	55697120	54670880	32915232	12516672	2991360	430080	33280	1024					201170880
11	7257600	92286720	348600384	640236608	681260800	455730880	199096128	57366144	10728960	1239040	78848	2048				2493884160
12	79833600	1044184320	4104206784	7943860672	9043446016	6594244800	3216982208	1069609728	242043648	36439040	3458048	184320	4096			33378497280
13	958003200	12849546240	52303200768	105949429248	127777331968	100161556352	53499736576	19944496000	5226743808	954809856	118317056	9398272	425984	8192		479753003520
14	12454041600	170876113920	717532245504	1512853934592	1918252650496	1599716134016	922196667392	377795420288	111383098112	23633465856	3561958400	369940480	24973312	974848	16384	7370651635200

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Technical details

Node id: n-d69109ad5c51d24c · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-feb6e39c62313058

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:     2,      2;
  2:     2,     10,       4;
  3:     4,     28,      36,       8;
  4:    12,    100,     188,     112,      16;
  5:    48,    448,    1040,     920,     320,      32;
  6:   240,   2432,    6496,    7240,    3760,     864,     64;
  7:  1440,  15552,   46032,   60032,   39760,   13664,   2240,    128;
  8: 10080, 114624,  367440,  538272,  421904,  186368,  45696,   5632,   256;
  9: 80640, 957312, 3273312, 5251168, 4668384, 2460192, 779520, 143616, 13824, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(4*n-8)*x+4*x^2)*P(n-2,x) + (2*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*P[n - 2, x] + (2*n - 3 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*v[i - 2] + (2*n - 3 + 4*x)*v[i - 1]); 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-ad2a1da0972eeb21
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-ad2a1da0972eeb21
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-ad2a1da0972eeb21

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-ad2a1da0972eeb21

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-6aa1e3cc7e408461 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.