A113214 → D

nonnegative coefficients

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

Recurrence

\(\left(2\,n + n x - 3\,n^{2} - n^{2} x + n^{3}\right)\,P_{n}(x) = \left(16 + 4\,x - 28\,n - 15\,n x - 2\,n x^{2} + 14\,n^{2} + 5\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - 12\,n x - 7\,n x^{2} - n x^{3} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	3	2						5
3	2	8	3					13
4	0	10	15	4				29
5	0	4	27	24	5			60
6	0	0	21	56	35	6		118
7	0	0	6	64	100	48	7	225

Show all 15 stored rows

Generated b-files: A023619 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-444982dca44ea50b · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-9c8169fbc8868e1d

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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 ?

A023619: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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: 0;
  1: 1;
  2: 3,  2;
  3: 2,  8,  3;
  4: 0, 10, 15,  4;
  5: 0,  4, 27, 24,   5;
  6: 0,  0, 21, 56,  35,   6;
  7: 0,  0,  6, 64, 100,  48,   7;
  8: 0,  0,  0, 36, 150, 162,  63,  8;
  9: 0,  0,  0,  8, 125, 300, 245, 80, 9;

%F Formula

The row polynomials satisfy (n^3-3*n^2+2*n-(n^2-n)*x)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(5*n^2-15*n+4)*x+2*n*x^2)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2-7*n+2)*x^2-n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 - (5*n^2 - 15*n + 4)*x + 2*n*x^2)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 - 7*n + 2)*x^2 - n*x^3)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n - (n^2 - n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (5*n^2 - 15*n + 4)*x + 2*n*x^2)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 - 7*n + 2)*x^2 - n*x^3)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n - (n^2 - n)*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-bba9fb27b4206840
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-bba9fb27b4206840
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-bba9fb27b4206840

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-bba9fb27b4206840

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A111125 e-7b745496cf172799
    
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    no recorded property covers this requirement
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-defa6799aaae8d2a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-f8dba443f4d27d49
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038279 e-4738d2d3e61fb87a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
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    preserver_image affine, shift 0 e-fbfad365d270ad64
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
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    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-1b92f65567f189e7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
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    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A007318 e-719615516772b132
    
    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-fa99d43ba99a1c05
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
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    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038243 e-94eaee4d49f2668d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
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    preserver_image affine, shift 0 e-32ad399df63f844d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027466 e-9b73612a2d328481
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-921246387fcf3a1f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-85e6814e89c84b8b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-e475554f8902db13
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038291 e-e67dd89b1bcbf6c7
    
    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-77c8319978ad9839
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027465 e-f9e5e2bc826d3dc0
    
    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-f73e26dac50fbf3d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	3	2						5
3	2	8	3					13
4	0	10	15	4				29
5	0	4	27	24	5			60
6	0	0	21	56	35	6		118
7	0	0	6	64	100	48	7	225

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	3	2						5
3	2	8	3					13
4	0	10	15	4				29
5	0	4	27	24	5			60
6	0	0	21	56	35	6		118
7	0	0	6	64	100	48	7	225

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	3	2						5
3	2	8	3					13
4	0	10	15	4				29
5	0	4	27	24	5			60
6	0	0	21	56	35	6		118
7	0	0	6	64	100	48	7	225