A064861 → D

nonnegative coefficients only rational roots

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

Recurrence

\(\left(4 + 4\,x + 13\,n + 28\,n x + 16\,n x^{2} + 9\,n^{2} + 24\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 14\,x + 8\,x^{2} + 53\,n + 283\,n x + 548\,n x^{2} + 448\,n x^{3} + 128\,n x^{4} + 27\,n^{2} + 204\,n^{2} x + 512\,n^{2} x^{2} + 528\,n^{2} x^{3} + 192\,n^{2} x^{4} + 27\,n^{3} x + 108\,n^{3} x^{2} + 144\,n^{3} x^{3} + 64\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-20 - 160\,x - 484\,x^{2} - 696\,x^{3} - 480\,x^{4} - 128\,x^{5} - 12\,n - 130\,n x - 482\,n x^{2} - 812\,n x^{3} - 640\,n x^{4} - 192\,n x^{5} - 18\,n^{2} x - 102\,n^{2} x^{2} - 212\,n^{2} x^{3} - 192\,n^{2} x^{4} - 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	3	4								7
1	5	16	12							33
2	6	26	36	16						84
3	8	50	114	112	40					324
4	9	66	189	264	180	48				756
5	11	102	387	768	840	480	112			2700
6	12	124	540	1284	1800	1488	672	128		6048
7	14	172	912	2724	5010	5808	4144	1664	288	20736

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2ff13446e75f9dd4 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-1788d34a8315ad53

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

OEIS matches ?

No OEIS matches.

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:  3,   4;
  1:  5,  16,   12;
  2:  6,  26,   36,   16;
  3:  8,  50,  114,  112,    40;
  4:  9,  66,  189,  264,   180,    48;
  5: 11, 102,  387,  768,   840,   480,   112;
  6: 12, 124,  540, 1284,  1800,  1488,   672,   128;
  7: 14, 172,  912, 2724,  5010,  5808,  4144,  1664,   288;
  8: 15, 200, 1170, 3940,  8415, 11820, 10920,  6400,  2160,  320;
  9: 17, 260, 1770, 7060, 18265, 32016, 38500, 31360, 16560, 5120, 704;

%F Formula

The row polynomials satisfy (9*n^2+13*n+4+(24*n^2+28*n+4)*x+(16*n^2+16*n)*x^2)*P(n,x) = (27*n^2+53*n+6+(27*n^3+204*n^2+283*n+14)*x+(108*n^3+512*n^2+548*n+8)*x^2+(144*n^3+528*n^2+448*n)*x^3+(64*n^3+192*n^2+128*n)*x^4)*P(n-1,x) - (12*n+20+(18*n^2+130*n+160)*x+(102*n^2+482*n+484)*x^2+(212*n^2+812*n+696)*x^3+(192*n^2+640*n+480)*x^4+(64*n^2+192*n+128)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 + 53*n + 6 + (27*n^3 + 204*n^2 + 283*n + 14)*x + (108*n^3 + 512*n^2 + 548*n + 8)*x^2 + (144*n^3 + 528*n^2 + 448*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*P[n - 1, x] - (12*n + 20 + (18*n^2 + 130*n + 160)*x + (102*n^2 + 482*n + 484)*x^2 + (212*n^2 + 812*n + 696)*x^3 + (192*n^2 + 640*n + 480)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(9*n^2 + 13*n + 4 + (24*n^2 + 28*n + 4)*x + (16*n^2 + 16*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] = 3 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 + 53*n + 6 + (27*n^3 + 204*n^2 + 283*n + 14)*x + (108*n^3 + 512*n^2 + 548*n + 8)*x^2 + (144*n^3 + 528*n^2 + 448*n)*x^3 + (64*n^3 + 192*n^2 + 128*n)*x^4)*v[i - 1] - (12*n + 20 + (18*n^2 + 130*n + 160)*x + (102*n^2 + 482*n + 484)*x^2 + (212*n^2 + 812*n + 696)*x^3 + (192*n^2 + 640*n + 480)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(9*n^2 + 13*n + 4 + (24*n^2 + 28*n + 4)*x + (16*n^2 + 16*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, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4ace2775516c13f5
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-4ace2775516c13f5
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-4ace2775516c13f5

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-4ace2775516c13f5

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.