A064861 → D+1

nonnegative coefficients

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

Recurrence

\(\left(34 + 112\,x + 144\,x^{2} + 64\,x^{3} + 80\,n + 176\,n x + 96\,n x^{2} + 24\,n^{2} + 32\,n^{2} x\right)\,P_{n}(x) = \left(148 + 926\,x + 2280\,x^{2} + 2656\,x^{3} + 1408\,x^{4} + 256\,x^{5} + 271\,n + 1490\,n x + 3148\,n x^{2} + 3136\,n x^{3} + 1472\,n x^{4} + 256\,n x^{5} + 30\,n^{2} + 352\,n^{2} x + 900\,n^{2} x^{2} + 768\,n^{2} x^{3} + 192\,n^{2} x^{4} - 27\,n^{3} - 72\,n^{3} x - 48\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-154 - 1194\,x - 3632\,x^{2} - 5520\,x^{3} - 4400\,x^{4} - 1728\,x^{5} - 256\,x^{6} + 18\,n - 26\,n x - 348\,n x^{2} - 592\,n x^{3} - 288\,n x^{4} + 36\,n^{2} + 204\,n^{2} x + 424\,n^{2} x^{2} + 384\,n^{2} x^{3} + 128\,n^{2} x^{4}\right)\,P_{n-1}'(x) + \left(-28 - 248\,x - 892\,x^{2} - 1664\,x^{3} - 1696\,x^{4} - 896\,x^{5} - 192\,x^{6} - 12\,n - 104\,n x - 348\,n x^{2} - 560\,n x^{3} - 432\,n x^{4} - 128\,n x^{5}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 4 + 7\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	7	2															13
1	6	21	20	4														51
2	7	32	49	28	4													120
3	9	58	139	150	68	8												432
4	10	75	222	327	246	84	8											972
5	12	113	438	897	1032	648	192	16										3348
6	13	136	602	1464	2121	1848	920	224	16									7344
7	15	186	998	3028	5691	6810	5112	2256	496	32								24624
8	16	215	1270	4330	9400	13503	12890	7960	2960	560	32							53136
9	18	277	1900	7650	20030	35669	43836	36860	20480	6960	1216	64						174960
10	19	312	2307	10140	29445	59352	84841	86196	61260	29280	8688	1344	64					373248
11	21	386	3225	16194	54435	129078	221523	277550	252492	163320	71952	19872	2880	128				1213056
12	22	427	3794	20433	74424	193641	370426	527887	561134	440412	249144	97104	23968	3136	128			2566080
13	24	513	5054	30415	124908	370335	817614	1366017	1736720	1674064	1206912	633696	231168	53760	6656	256		8258112
14	25	560	5812	37072	162526	518784	1245748	2292176	3259921	3586288	3030832	1936704	909664	299264	63232	7168	256	17356032

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Technical details

Node id: n-afa83802600b0da9 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7c9cde47112aa257

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:  4,   7,    2;
  1:  6,  21,   20,    4;
  2:  7,  32,   49,   28,     4;
  3:  9,  58,  139,  150,    68,     8;
  4: 10,  75,  222,  327,   246,    84,     8;
  5: 12, 113,  438,  897,  1032,   648,   192,    16;
  6: 13, 136,  602, 1464,  2121,  1848,   920,   224,    16;
  7: 15, 186,  998, 3028,  5691,  6810,  5112,  2256,   496,   32;
  8: 16, 215, 1270, 4330,  9400, 13503, 12890,  7960,  2960,  560,   32;
  9: 18, 277, 1900, 7650, 20030, 35669, 43836, 36860, 20480, 6960, 1216, 64;

%F Formula

The row polynomials satisfy (24*n^2+80*n+34+(32*n^2+176*n+112)*x+(96*n+144)*x^2+64*x^3)*P(n,x) = -(27*n^3-30*n^2-271*n-148+(72*n^3-352*n^2-1490*n-926)*x+(48*n^3-900*n^2-3148*n-2280)*x^2-(768*n^2+3136*n+2656)*x^3-(192*n^2+1472*n+1408)*x^4-(256*n+256)*x^5)*P(n-1,x) + (36*n^2+18*n-154+(204*n^2-26*n-1194)*x+(424*n^2-348*n-3632)*x^2+(384*n^2-592*n-5520)*x^3+(128*n^2-288*n-4400)*x^4-1728*x^5-256*x^6)*d/dx P(n-1,x) - (12*n+28+(104*n+248)*x+(348*n+892)*x^2+(560*n+1664)*x^3+(432*n+1696)*x^4+(128*n+896)*x^5+192*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4 + 7*x + 2*x^2.

%t Mathematica

P[0, x_] := 4 + 7*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 30*n^2 - 271*n - 148 + (72*n^3 - 352*n^2 - 1490*n - 926)*x + (48*n^3 - 900*n^2 - 3148*n - 2280)*x^2 - (768*n^2 + 3136*n + 2656)*x^3 - (192*n^2 + 1472*n + 1408)*x^4 - (256*n + 256)*x^5)*P[n - 1, x] + (36*n^2 + 18*n - 154 + (204*n^2 - 26*n - 1194)*x + (424*n^2 - 348*n - 3632)*x^2 + (384*n^2 - 592*n - 5520)*x^3 + (128*n^2 - 288*n - 4400)*x^4 - 1728*x^5 - 256*x^6)*D[P[n - 1, x], x] - (12*n + 28 + (104*n + 248)*x + (348*n + 892)*x^2 + (560*n + 1664)*x^3 + (432*n + 1696)*x^4 + (128*n + 896)*x^5 + 192*x^6)*D[P[n - 1, x], {x, 2}])/(24*n^2 + 80*n + 34 + (32*n^2 + 176*n + 112)*x + (96*n + 144)*x^2 + 64*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 7*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(27*n^3 - 30*n^2 - 271*n - 148 + (72*n^3 - 352*n^2 - 1490*n - 926)*x + (48*n^3 - 900*n^2 - 3148*n - 2280)*x^2 - (768*n^2 + 3136*n + 2656)*x^3 - (192*n^2 + 1472*n + 1408)*x^4 - (256*n + 256)*x^5)*v[i - 1] + (36*n^2 + 18*n - 154 + (204*n^2 - 26*n - 1194)*x + (424*n^2 - 348*n - 3632)*x^2 + (384*n^2 - 592*n - 5520)*x^3 + (128*n^2 - 288*n - 4400)*x^4 - 1728*x^5 - 256*x^6)*deriv(v[i - 1], x) - (12*n + 28 + (104*n + 248)*x + (348*n + 892)*x^2 + (560*n + 1664)*x^3 + (432*n + 1696)*x^4 + (128*n + 896)*x^5 + 192*x^6)*deriv(deriv(v[i - 1], x), x))/(24*n^2 + 80*n + 34 + (32*n^2 + 176*n + 112)*x + (96*n + 144)*x^2 + 64*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-fb87f78803cc76fe
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-fb87f78803cc76fe
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-fb87f78803cc76fe

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-fb87f78803cc76fe

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.