A232535 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-3\,x - 9\,x^{2} + n + n x + 15\,n x^{2} + n^{2} + 3\,n^{2} x - 6\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(3\,x^{2} - 21\,x^{3} + 2\,n x - 2\,n x^{2} + 12\,n x^{3}\right)\,P_{n-1}'(x) + \left(6\,x^{3} - 6\,x^{4}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	3	8	1													12
3	4	25	18	1												48
4	5	56	98	32	1											192
5	6	105	336	270	50	1										768
6	7	176	891	1320	605	72	1									3072
7	8	273	2002	4719	4004	1183	98	1								12288
8	9	400	4004	13728	18590	10192	2100	128	1							49152
9	10	561	7344	34476	68068	59670	22848	3468	162	1						196608
10	11	760	12597	77520	209950	268736	164730	46512	5415	200	1					786432
11	12	1001	20482	159885	568480	999362	895356	405042	87780	8085	242	1				3145728
12	13	1288	31878	307648	1389223	3209328	3952228	2615008	908523	155848	11638	288	1			12582912
13	14	1625	47840	559130	3124550	9174815	14858000	13520780	6874010	1891175	263120	16250	338	1		50331648
14	15	2016	69615	968760	6561555	23860200	49142835	58837680	41158845	16576560	3700125	425880	22113	392	1	201326592

Generated b-files: A002001 (row sums), A164346 (row sums), A232535 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-54f319533dce3c44 · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-d583a30bb4254b7d

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 ?

A092898: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A002001: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A232535: reversed triangle, 58 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A164346: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A110594: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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,   1;
  2:  3,   8,    1;
  3:  4,  25,   18,     1;
  4:  5,  56,   98,    32,     1;
  5:  6, 105,  336,   270,    50,     1;
  6:  7, 176,  891,  1320,   605,    72,     1;
  7:  8, 273, 2002,  4719,  4004,  1183,    98,    1;
  8:  9, 400, 4004, 13728, 18590, 10192,  2100,  128,   1;
  9: 10, 561, 7344, 34476, 68068, 59670, 22848, 3468, 162, 1;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (n^2+n+(3*n^2+n-3)*x-(6*n^2-15*n+9)*x^2)*P(n-1,x) + (2*n*x-(2*n-3)*x^2+(12*n-21)*x^3)*d/dx P(n-1,x) + (6*x^3-6*x^4)*d^2/dx^2 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[((n^2 + n + (3*n^2 + n - 3)*x - (6*n^2 - 15*n + 9)*x^2)*P[n - 1, x] + (2*n*x - (2*n - 3)*x^2 + (12*n - 21)*x^3)*D[P[n - 1, x], x] + (6*x^3 - 6*x^4)*D[P[n - 1, x], {x, 2}])/(n^2)]];
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] = ((n^2 + n + (3*n^2 + n - 3)*x - (6*n^2 - 15*n + 9)*x^2)*v[i - 1] + (2*n*x - (2*n - 3)*x^2 + (12*n - 21)*x^3)*deriv(v[i - 1], x) + (6*x^3 - 6*x^4)*deriv(deriv(v[i - 1], x), x))/(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.

A232535: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   8,    3;
  3: 1,  18,   25,     4;
  4: 1,  32,   98,    56,     5;
  5: 1,  50,  270,   336,   105,     6;
  6: 1,  72,  605,  1320,   891,   176,     7;
  7: 1,  98, 1183,  4004,  4719,  2002,   273,    8;
  8: 1, 128, 2100, 10192, 18590, 13728,  4004,  400,   9;
  9: 1, 162, 3468, 22848, 59670, 68068, 34476, 7344, 561, 10;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n^2*P(n,x) = (n^2+n+(3*n^2+n-3)*x-(6*n^2-15*n+9)*x^2)*P(n-1,x) + (2*n*x-(2*n-3)*x^2+(12*n-21)*x^3)*d/dx P(n-1,x) + (6*x^3-6*x^4)*d^2/dx^2 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[((n^2 + n + (3*n^2 + n - 3)*x - (6*n^2 - 15*n + 9)*x^2)*P[n - 1, x] + (2*n*x - (2*n - 3)*x^2 + (12*n - 21)*x^3)*D[P[n - 1, x], x] + (6*x^3 - 6*x^4)*D[P[n - 1, x], {x, 2}])/(n^2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = ((n^2 + n + (3*n^2 + n - 3)*x - (6*n^2 - 15*n + 9)*x^2)*v[i - 1] + (2*n*x - (2*n - 3)*x^2 + (12*n - 21)*x^3)*deriv(v[i - 1], x) + (6*x^3 - 6*x^4)*deriv(deriv(v[i - 1], x), x))/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-3408e49d5219d076
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-3408e49d5219d076
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-3408e49d5219d076

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 reversal, shift 0 e-3408e49d5219d076

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-a75bd3d9d36c6f35 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.

A232535: Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.

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