A152842 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11 - 58\,x - 36\,x^{2} + 36\,x^{3} - 11\,n + 18\,n x + 54\,n x^{2} + 12\,n^{2} + 18\,n^{2} x\right)\,P_{n}(x) = \left(-26 - 18\,x + 106\,x^{2} - 282\,x^{3} - 576\,x^{4} - 216\,x^{5} - 106\,n - 550\,n x - 931\,n x^{2} - 132\,n x^{3} + 522\,n x^{4} + 216\,n x^{5} + 108\,n^{2} + 498\,n^{2} x + 867\,n^{2} x^{2} + 612\,n^{2} x^{3} + 162\,n^{2} x^{4} - 16\,n^{3} - 48\,n^{3} x - 36\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-15 - 122\,x - 543\,x^{2} - 1546\,x^{3} - 2262\,x^{4} - 1368\,x^{5} - 216\,x^{6} - 56\,n - 456\,n x - 1324\,n x^{2} - 1608\,n x^{3} - 684\,n x^{4} + 16\,n^{2} + 112\,n^{2} x + 276\,n^{2} x^{2} + 288\,n^{2} x^{3} + 108\,n^{2} x^{4}\right)\,P_{n-1}'(x) + \left(-1 - 12\,x - 72\,x^{2} - 256\,x^{3} - 501\,x^{4} - 468\,x^{5} - 162\,x^{6} - 4\,n - 44\,n x - 184\,n x^{2} - 360\,n x^{3} - 324\,n x^{4} - 108\,n x^{5}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	5	10	3													18
3	8	37	42	9												96
4	9	52	94	60	9											224
5	12	103	316	414	216	27										1088
6	13	126	465	820	711	270	27									2432
7	16	201	1014	2623	3684	2727	918	81								11264
8	17	232	1308	3944	6886	7032	3996	1080	81							24576
9	20	331	2328	9068	21376	31290	28008	14364	3564	243						110592
10	21	370	2815	12120	32530	56524	63870	45720	19305	4050	243					237568
11	24	493	4450	23205	77320	171946	258732	261090	170640	66825	13122	729				1048576
12	25	540	5178	29060	105975	263640	456940	553032	460215	253260	85050	14580	729			2228224
13	28	687	7572	49514	213720	641037	1369768	2102268	2304396	1766745	907524	287226	46656	2187		9699328
14	29	742	8589	59500	275009	895146	2110913	3653416	4646775	4300506	2828007	1263276	352107	51030	2187	20447232

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Technical details

Node id: n-7fa0a8eeaf34c691 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2a74beb062736e56

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 ?

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:  1;
  1:  4,   3;
  2:  5,  10,    3;
  3:  8,  37,   42,    9;
  4:  9,  52,   94,   60,     9;
  5: 12, 103,  316,  414,   216,    27;
  6: 13, 126,  465,  820,   711,   270,    27;
  7: 16, 201, 1014, 2623,  3684,  2727,   918,    81;
  8: 17, 232, 1308, 3944,  6886,  7032,  3996,  1080,   81;
  9: 20, 331, 2328, 9068, 21376, 31290, 28008, 14364, 3564, 243;

%F Formula

The row polynomials satisfy (12*n^2-11*n-11+(18*n^2+18*n-58)*x+(54*n-36)*x^2+36*x^3)*P(n,x) = -(16*n^3-108*n^2+106*n+26+(48*n^3-498*n^2+550*n+18)*x+(36*n^3-867*n^2+931*n-106)*x^2-(612*n^2-132*n-282)*x^3-(162*n^2+522*n-576)*x^4-(216*n-216)*x^5)*P(n-1,x) + (16*n^2-56*n-15+(112*n^2-456*n-122)*x+(276*n^2-1324*n-543)*x^2+(288*n^2-1608*n-1546)*x^3+(108*n^2-684*n-2262)*x^4-1368*x^5-216*x^6)*d/dx P(n-1,x) - (4*n+1+(44*n+12)*x+(184*n+72)*x^2+(360*n+256)*x^3+(324*n+501)*x^4+(108*n+468)*x^5+162*x^6)*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[(-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*P[n - 1, x] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*D[P[n - 1, x], x] - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*D[P[n - 1, x], {x, 2}])/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*x^3)]];
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] = (-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*v[i - 1] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*deriv(v[i - 1], x) - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*deriv(deriv(v[i - 1], x), x))/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*x^3)); 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-c18e1ce17600bec1
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-c18e1ce17600bec1
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-c18e1ce17600bec1

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-c18e1ce17600bec1

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 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.