A257619 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-25 + 18\,x + 9\,n\right)\,P_{n}(x) = \left(352 - 2284\,x + 11116\,x^{2} - 12420\,x^{3} + 2592\,x^{4} - 198\,n + 1638\,n x - 12906\,n x^{2} + 10692\,n x^{3} - 1458\,n x^{4} - 324\,n^{2} x + 5346\,n^{2} x^{2} - 2187\,n^{2} x^{3} - 729\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(288\,x - 4212\,x^{2} + 9270\,x^{3} - 6804\,x^{4} + 1458\,x^{5} - 162\,n x + 3483\,n x^{2} - 5508\,n x^{3} + 2187\,n x^{4} - 729\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-307 + 1259\,x - 1044\,x^{2} + 162\,x^{3} + 117\,n - 819\,n x + 405\,n x^{2} + 162\,n^{2} x\right)\,P_{n-1}(x) + \left(-225\,x + 387\,x^{2} - 162\,x^{3} + 81\,n x - 81\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 2\,x\)

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	2														6
2	48	52	4													104
3	572	1692	588	8												2860
4	6452	51556	41868	6500	16											106392
5	71436	1347092	2551376	923460	71564	32										4964960
6	787012	31727412	126455088	98875880	19404972	787332	64									278037760
7	8660140	701442404	5328558468	8303612120	3392678404	398351268	8660908	128								18141963840
8	95268708	14920837636	200736794116	569300953404	448421586764	108918044060	8079663756	95270500	256							1350568419200
9	1047972428	309944254356	7006192228872	33583551401436	46949455058328	21528397351692	3365051684424	162831916548	1047976524	512						112907519845120
10	11527734596	6344553225940	231984478825992	1777255742048928	4102953084574608	3287016092613960	959154359191800	101604757329408	3270262348332	11527743812	1024					10469606385638400
11	126805165548	128700918783812	7402973902777436	86914476636420816	313038121696581624	408689483684496504	206203645408516920	40686893511187344	3024812459947164	65555108445156	126805186028	2048				1066154916937510400
12	1394857009444	2596209279495588	230126802890025084	4010153646163437076	21566804229273301992	43350857420519465928	35355700646606554872	11984421348682155720	1669957482270634164	89250537338711188	1312750638303948	1394857054500	4096			118261183863376153600
13	15343427517580	52193392992394484	7020225889770714624	177142498285727136052	1375003160969205004820	4068874865370200256360	5070153052190527743552	2763849741342251557512	659585094721449300756	67016569382643668260	2618358227602958720	26273145912555204	15343427615884	8192		14191342063605138432000
14	168777703594500	1047105323053345396	211258411866414865888	7571014971204254425752	82582964041481476674244	347307456345721318730140	631924226692826754237408	523944523147329863099664	200468305785318027927756	34893479283811831025772	2647887290678631876160	76526600470011931096	525662382821399148	168777703807492	16384	1831629215675969866956800

Technical details

Node id: n-939a2782d4f5adb9 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2f50bf37151e49a3

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:          1;
  1:          4,            2;
  2:         48,           52,             4;
  3:        572,         1692,           588,              8;
  4:       6452,        51556,         41868,           6500,             16;
  5:      71436,      1347092,       2551376,         923460,          71564,             32;
  6:     787012,     31727412,     126455088,       98875880,       19404972,         787332,            64;
  7:    8660140,    701442404,    5328558468,     8303612120,     3392678404,      398351268,       8660908,          128;
  8:   95268708,  14920837636,  200736794116,   569300953404,   448421586764,   108918044060,    8079663756,     95270500,        256;
  9: 1047972428, 309944254356, 7006192228872, 33583551401436, 46949455058328, 21528397351692, 3365051684424, 162831916548, 1047976524, 512;

%F Formula

The row polynomials satisfy (9*n-25+18*x)*P(n,x) = -(198*n-352+(324*n^2-1638*n+2284)*x+(729*n^3-5346*n^2+12906*n-11116)*x^2+(2187*n^2-10692*n+12420)*x^3+(1458*n-2592)*x^4)*P(n-2,x) - ((162*n-288)*x+(729*n^2-3483*n+4212)*x^2-(729*n^2-5508*n+9270)*x^3-(2187*n-6804)*x^4-1458*x^5)*d/dx P(n-2,x) + (117*n-307+(162*n^2-819*n+1259)*x+(405*n-1044)*x^2+162*x^3)*P(n-1,x) + ((81*n-225)*x-(81*n-387)*x^2-162*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*P[n - 2, x] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*D[P[n - 2, x], x] + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*P[n - 1, x] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*D[P[n - 1, x], x])/(9*n - 25 + 18*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(198*n - 352 + (324*n^2 - 1638*n + 2284)*x + (729*n^3 - 5346*n^2 + 12906*n - 11116)*x^2 + (2187*n^2 - 10692*n + 12420)*x^3 + (1458*n - 2592)*x^4)*v[i - 2] - ((162*n - 288)*x + (729*n^2 - 3483*n + 4212)*x^2 - (729*n^2 - 5508*n + 9270)*x^3 - (2187*n - 6804)*x^4 - 1458*x^5)*deriv(v[i - 2], x) + (117*n - 307 + (162*n^2 - 819*n + 1259)*x + (405*n - 1044)*x^2 + 162*x^3)*v[i - 1] + ((81*n - 225)*x - (81*n - 387)*x^2 - 162*x^3)*deriv(v[i - 1], x))/(9*n - 25 + 18*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.?
    
    ↑ Liu–Wang theorem (above)

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