A133214 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-1 - 8\,x - 16\,x^{2} - n - 8\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(9 + 12\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 13 + 32\,x + 16\,x^{2}\)

\(P_{1}(x) = 63 + 228\,x + 240\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	13	32	16								61
1	63	228	240	64							595
2	321	1536	2496	1536	256						6145
3	1683	10020	22080	21760	8960	1024					65527
4	8989	64032	178320	243200	165120	49152	4096				712909
5	48639	403396	1358448	2372160	2266880	1139712	258048	16384			7863667
6	265729	2514944	9936640	21174272	26199040	18808832	7340032	1310720	65536		87615745
7	1462563	15554628	70541568	177558528	270240768	253661184	143130624	44826624	6488064	262144	983726695

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e1b5185a7301abc6 · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-e88a2f797c4f541a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A098659: 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:       13,        32,         16;
  1:       63,       228,        240,          64;
  2:      321,      1536,       2496,        1536,         256;
  3:     1683,     10020,      22080,       21760,        8960,        1024;
  4:     8989,     64032,     178320,      243200,      165120,       49152,        4096;
  5:    48639,    403396,    1358448,     2372160,     2266880,     1139712,      258048,       16384;
  6:   265729,   2514944,    9936640,    21174272,    26199040,    18808832,     7340032,     1310720,      65536;
  7:  1462563,  15554628,   70541568,   177558528,   270240768,   253661184,   143130624,    44826624,    6488064,     262144;
  8:  8097453,  95600160,  489425040,  1421383680,  2570695680,  2989228032,  2224373760,  1018429440,  262471680,   31457280,   1048576;
  9: 45046719, 584585892, 3334581360, 10977384000, 23014210560, 31972233216, 29660725248, 18047631360, 6866534400, 1485045760, 149946368, 4194304;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(n+1+(8*n+8)*x+(16*n+16)*x^2)*P(n-2,x) + (6*n+9+(8*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 32*x + 16*x^2, P(1,x) = 63 + 228*x + 240*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 13 + 32*x + 16*x^2;
P[1, x_] := 63 + 228*x + 240*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (8*n + 8)*x + (16*n + 16)*x^2)*P[n - 2, x] + (6*n + 9 + (8*n + 12)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 32*x + 16*x^2; v[2] = 63 + 228*x + 240*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (8*n + 8)*x + (16*n + 16)*x^2)*v[i - 2] + (6*n + 9 + (8*n + 12)*x)*v[i - 1])/(n + 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-3165932d1e38479b
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-3165932d1e38479b
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-3165932d1e38479b
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-3165932d1e38479b

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

basis_map A013609 e-3165932d1e38479b

Geometric basis map (elementary) polytool ?
- premise (sequence): Delannoy paths counted by number of weak peaks.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4ab8ec5aa6b36e54 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A104684 → basis A013611

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023