A169803 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	1					1
1	2	1				3
2	3	2				5
3	4	5	1			10
4	5	10	3			18
5	6	17	9	1		33
6	7	26	22	4		59
7	8	37	45	14	1	105

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-c7a51edb27425df9 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-908e152c6e14648f

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 ?

A010049: 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:  2,  1;
  2:  3,  2;
  3:  4,  5,   1;
  4:  5, 10,   3;
  5:  6, 17,   9,  1;
  6:  7, 26,  22,  4;
  7:  8, 37,  45, 14,  1;
  8:  9, 50,  81, 40,  5;
  9: 10, 65, 133, 95, 20, 1;

%F Formula

The row polynomials satisfy (n^3+n^2+(4*n^2-n-5)*x+(4*n+4)*x^2)*P(n,x) = (n^3+2*n^2+n+(2*n^3+4*n^2-3*n-5)*x+(8*n^2+2*n+4)*x^2+8*n*x^3)*P(n-1,x) - ((n^2+2*n+1)*x+(4*n^2+12*n+7)*x^2+(16*n+16)*x^3+16*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[((n^3 + 2*n^2 + n + (2*n^3 + 4*n^2 - 3*n - 5)*x + (8*n^2 + 2*n + 4)*x^2 + 8*n*x^3)*P[n - 1, x] - ((n^2 + 2*n + 1)*x + (4*n^2 + 12*n + 7)*x^2 + (16*n + 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 + (4*n^2 - n - 5)*x + (4*n + 4)*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] = ((n^3 + 2*n^2 + n + (2*n^3 + 4*n^2 - 3*n - 5)*x + (8*n^2 + 2*n + 4)*x^2 + 8*n*x^3)*v[i - 1] - ((n^2 + 2*n + 1)*x + (4*n^2 + 12*n + 7)*x^2 + (16*n + 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 + (4*n^2 - n - 5)*x + (4*n + 4)*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, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c190684bc410f050
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-c190684bc410f050
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-c190684bc410f050

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-c190684bc410f050

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-2d7abb76d58e6037
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    basis_map A027466 e-1781530cbfe1e9bf
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-742e5241e9a4e627
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
    
    basis_map A027467 e-361937c2693b166b
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-8f58f94d7421b40d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    
    basis_map A027465 e-5c2b8bc6b43c730e
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-a2e7017ebd9e3749
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    
    basis_map A007318 e-ba7f4c6c74dd05c0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-51281bad32c8f5cd
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
    
    preserver_image reversal, shift 1 e-fb809a8e7c094d2b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-266d85ca03a298fa
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
    
    basis_map A008288 e-e771e35dc9ee3fbd
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-33dba7e79fe9caa8
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
    
    ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
    
    basis_map A030528 e-be49b310b519d57a
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ theorem_check ma_wang_interval e-f811a014585c6352 (expanded above)
  - parent_provers paper_theorem e-4d4f59f3aacb50cf root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

References

[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
[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.