basis-map coefficients of A132812 → D+1

Proved for all n: all roots real what this means

Recurrence

\(\left(12\,n - 10\,n x + 2\,n x^{2} - n^{2} x + n^{2} x^{2} - n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(12 - 10\,x + 2\,x^{2} + 6\,n + 11\,n x - 12\,n x^{2} + 3\,n x^{3} + 2\,n^{2} + 10\,n^{2} x - 8\,n^{2} x^{2} + 3\,n^{2} x^{3} + 6\,n^{3} - 8\,n^{3} x + 6\,n^{3} x^{2} - 2\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-12 + 30\,x - 28\,x^{2} + 12\,x^{3} - 2\,x^{4} - 6\,n + 13\,n x - 13\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 2\,n^{2} - 11\,n^{2} x + 13\,n^{2} x^{2} - 4\,n^{2} x^{3} - 4\,n^{3} + 6\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	0	9	3						12
3	-12	12	24	4					28
4	-10	-60	70	50	5				55
5	60	-180	-120	240	90	6			96
6	70	315	-1050	35	630	147	7		154
7	-280	1400	0	-3640	1120	1400	224	8	232

Show all 15 stored rows

Generated b-files: A006000 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-100368592cf6d595 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-daf4cab55006ac12

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): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A006000: 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 ?

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,     2;
  2:    0,     9,     3;
  3:  -12,    12,    24,     4;
  4:  -10,   -60,    70,    50,      5;
  5:   60,  -180,  -120,   240,     90,      6;
  6:   70,   315, -1050,    35,    630,    147,     7;
  7: -280,  1400,     0, -3640,   1120,   1400,   224,    8;
  8: -378, -1512,  9324, -6300,  -8694,   4788,  2772,  324,   9;
  9: 1260, -8820,  5040, 33600, -36540, -13860, 14280, 5040, 450, 10;

%F Formula

The row polynomials satisfy (n^4-n^3+12*n+(2*n^3-n^2-10*n)*x+(n^2+2*n)*x^2)*P(n,x) = -(2*n^4-6*n^3-2*n^2-6*n-12-(3*n^4-8*n^3+10*n^2+11*n-10)*x-(6*n^3-8*n^2-12*n+2)*x^2-(3*n^2+3*n)*x^3)*P(n-1,x) - (4*n^3-2*n^2+6*n+12-(6*n^3-11*n^2+13*n+30)*x+(2*n^3-13*n^2+13*n+28)*x^2+(4*n^2-8*n-12)*x^3+(2*n+2)*x^4)*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[(-(2*n^4 - 6*n^3 - 2*n^2 - 6*n - 12 - (3*n^4 - 8*n^3 + 10*n^2 + 11*n - 10)*x - (6*n^3 - 8*n^2 - 12*n + 2)*x^2 - (3*n^2 + 3*n)*x^3)*P[n - 1, x] - (4*n^3 - 2*n^2 + 6*n + 12 - (6*n^3 - 11*n^2 + 13*n + 30)*x + (2*n^3 - 13*n^2 + 13*n + 28)*x^2 + (4*n^2 - 8*n - 12)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 - n^3 + 12*n + (2*n^3 - n^2 - 10*n)*x + (n^2 + 2*n)*x^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] = (-(2*n^4 - 6*n^3 - 2*n^2 - 6*n - 12 - (3*n^4 - 8*n^3 + 10*n^2 + 11*n - 10)*x - (6*n^3 - 8*n^2 - 12*n + 2)*x^2 - (3*n^2 + 3*n)*x^3)*v[i - 1] - (4*n^3 - 2*n^2 + 6*n + 12 - (6*n^3 - 11*n^2 + 13*n + 30)*x + (2*n^3 - 13*n^2 + 13*n + 28)*x^2 + (4*n^2 - 8*n - 12)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 - n^3 + 12*n + (2*n^3 - n^2 - 10*n)*x + (n^2 + 2*n)*x^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, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-908b120a7c606a06

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-908b120a7c606a06

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - preserver_image affine, shift 0 e-676bb32d6a1b9c64
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05
    
    Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-29ef477df4172e22
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
    
    basis_map A007318 e-a32917d44ac940a7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    no recorded property covers this requirement
    
    preserver_image reversal, shift 0 e-fcd77baa2d5e1258
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
    
    ↵ basis_map A007318 e-a32917d44ac940a7 (expanded above)
    
    theorem_check liu_wang e-f8a41d41829f0ed1 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)
  - theorem_check three_term_recurrence e-72ab2c9b35ff70ef 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.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[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