A172431 → 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

\(\left(2\,n + 4\,n^{2} + 4\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(-4\,x - 4\,x^{2} + 2\,n + 4\,n x - 8\,n x^{2} + 4\,n^{2} + 28\,n^{2} x + 36\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 8\,x^{3} - 2\,n x - 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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	4														5
2	1	8	9													18
3	1	12	30	16												59
4	1	16	63	80	25											185
5	1	20	108	224	175	36										564
6	1	24	165	480	630	336	49									1685
7	1	28	234	880	1650	1512	588	64								4957
8	1	32	315	1456	3575	4752	3234	960	81							14406
9	1	36	408	2240	6825	12012	12012	6336	1485	100						41455
10	1	40	513	3264	11900	26208	35035	27456	11583	2200	121					118321
11	1	44	630	4560	19380	51408	86632	91520	57915	20020	3146	144				335400
12	1	48	759	6160	29925	93024	189924	254592	218790	114400	33033	4368	169			945193
13	1	52	900	8096	44275	158004	379848	620160	680238	486200	213928	52416	5915	196		2650229
14	1	56	1053	10400	63250	255024	706629	1364352	1831410	1679600	1016158	381888	80444	7840	225	7398330

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Technical details

Node id: n-68ad8eba8fb6df33 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7722c8666024957

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 ?

A377866: 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: 1;
  1: 1,  4;
  2: 1,  8,   9;
  3: 1, 12,  30,   16;
  4: 1, 16,  63,   80,   25;
  5: 1, 20, 108,  224,  175,    36;
  6: 1, 24, 165,  480,  630,   336,    49;
  7: 1, 28, 234,  880, 1650,  1512,   588,   64;
  8: 1, 32, 315, 1456, 3575,  4752,  3234,  960,   81;
  9: 1, 36, 408, 2240, 6825, 12012, 12012, 6336, 1485, 100;

%F Formula

The row polynomials satisfy (4*n^2+2*n+(8*n^3+4*n^2)*x)*P(n,x) = (4*n^2+2*n+(8*n^3+28*n^2+4*n-4)*x+(24*n^3+36*n^2-8*n-4)*x^2)*P(n-1,x) - ((2*n+1)*x+(4*n^2+16*n+6)*x^2+(16*n^2+32*n+8)*x^3)*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[((4*n^2 + 2*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (24*n^3 + 36*n^2 - 8*n - 4)*x^2)*P[n - 1, x] - ((2*n + 1)*x + (4*n^2 + 16*n + 6)*x^2 + (16*n^2 + 32*n + 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*x)]];
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] = ((4*n^2 + 2*n + (8*n^3 + 28*n^2 + 4*n - 4)*x + (24*n^3 + 36*n^2 - 8*n - 4)*x^2)*v[i - 1] - ((2*n + 1)*x + (4*n^2 + 16*n + 6)*x^2 + (16*n^2 + 32*n + 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*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-22126d71e196a99d
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-22126d71e196a99d
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-22126d71e196a99d
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-22126d71e196a99d

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-22126d71e196a99d

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Even row Pascal-square read by antidiagonals.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-83282836b5ac538e 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