A290316 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-17 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(44 - 1311\,x + 32481\,x^{2} - 22086\,x^{3} + 2592\,x^{4} - 24\,n + 1098\,n x - 38934\,n x^{2} + 16038\,n x^{3} - 972\,n x^{4} - 216\,n^{2} x + 15228\,n^{2} x^{2} - 2916\,n^{2} x^{3} - 1944\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(33\,x - 1800\,x^{2} + 6318\,x^{3} - 3807\,x^{4} + 486\,x^{5} - 18\,n x + 1566\,n x^{2} - 5022\,n x^{3} + 1458\,n x^{4} - 324\,n^{2} x^{2} + 972\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-79 + 1407\,x - 693\,x^{2} + 54\,x^{3} + 30\,n - 1098\,n x + 270\,n x^{2} + 216\,n^{2} x\right)\,P_{n-1}(x) + \left(-51\,x + 171\,x^{2} - 54\,x^{3} + 18\,n x - 54\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 7 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	6														13
3	49	228	90													367
4	235	4662	8694	2160												15751
5	997	55860	371196	399708	71280											899041
6	4063	525558	8636148	29874312	21739428	2993760										63773269
7	16345	4373604	146947662	1170865800	2551341564	1370228400	152681760									5396455135
8	65491	33998550	2092102290	32078842560	153632231820	234575876232	98362803600	9160905600								529936826143
9	262093	253904052	26680731840	707403008340	6273660789540	20342762406576	23291048663784	7928804037600	632102486400							59202716290225
10	1048519	1850452182	316684918512	13541266947960	200486987121960	1171238676374808	2772692349214104	2494472439737664	709401128349600	49303993939200						7411455378104509
11	4194241	13283448612	3580357618386	235222336117440	5429497748474520	51674736907265472	215813813133273264	392845626889816752	287361806607076320	69778844115494400	4289447472710400					1027432588855489807
12	16777147	94456318710	39130771784526	3813060323037168	130766678354105640	1896145077030788304	12666548529261567360	39997232895684444192	58136839494796543536	35486961877943225280	7485488188617196800	411786957380198400				156215621984635987063
13	67108789	667686986676	417517962172740	58763201800078476	2889729578077069896	60877488506314239648	609138951907895397072	3032286511482392313600	7539846154605962134992	9005375798369513900352	4681475374206136703040	869817131797568371200	43237630524920832000			25845003952366297308481
14	268435375	4702195924278	4378484168716140	872040416057117928	59848934760536441148	1768338880551010753776	25291968172849132228656	186101825876667608509440	720874582977125090948976	1455563125825623910490976	1460941462288285051162752	657508260147188960666880	108846725323672505433600	4929089879840974848000		4621886104730167471677925
15	1073741737	33038583306852	45343592658304470	12575493019700789112	1179239060119536930852	47560781328046742049552	939012252897837293479584	9753268131876142642785120	55246428646409798685106704	172046539087524878370767424	288961152970527162135572256	248160162491902877117402880	98002673388301080151975680	14594505877197455560243200	606278055220439906304000	888358773543115490158759423

Technical details

Node id: n-75904bb0bb525186 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cb98ad15abc25bd9

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 -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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       1;
   2:       7,          6;
   3:      49,        228,           90;
   4:     235,       4662,         8694,           2160;
   5:     997,      55860,       371196,         399708,           71280;
   6:    4063,     525558,      8636148,       29874312,        21739428,          2993760;
   7:   16345,    4373604,    146947662,     1170865800,      2551341564,       1370228400,        152681760;
   8:   65491,   33998550,   2092102290,    32078842560,    153632231820,     234575876232,      98362803600,       9160905600;
   9:  262093,  253904052,  26680731840,   707403008340,   6273660789540,   20342762406576,   23291048663784,    7928804037600,    632102486400;
  10: 1048519, 1850452182, 316684918512, 13541266947960, 200486987121960, 1171238676374808, 2772692349214104, 2494472439737664, 709401128349600, 49303993939200;

%F Formula

The row polynomials satisfy (6*n-17+6*x)*P(n,x) = -(24*n-44+(216*n^2-1098*n+1311)*x+(1944*n^3-15228*n^2+38934*n-32481)*x^2+(2916*n^2-16038*n+22086)*x^3+(972*n-2592)*x^4)*P(n-2,x) - ((18*n-33)*x+(324*n^2-1566*n+1800)*x^2-(972*n^2-5022*n+6318)*x^3-(1458*n-3807)*x^4-486*x^5)*d/dx P(n-2,x) + (30*n-79+(216*n^2-1098*n+1407)*x+(270*n-693)*x^2+54*x^3)*P(n-1,x) + ((18*n-51)*x-(54*n-171)*x^2-54*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 7 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 44 + (216*n^2 - 1098*n + 1311)*x + (1944*n^3 - 15228*n^2 + 38934*n - 32481)*x^2 + (2916*n^2 - 16038*n + 22086)*x^3 + (972*n - 2592)*x^4)*P[n - 2, x] - ((18*n - 33)*x + (324*n^2 - 1566*n + 1800)*x^2 - (972*n^2 - 5022*n + 6318)*x^3 - (1458*n - 3807)*x^4 - 486*x^5)*D[P[n - 2, x], x] + (30*n - 79 + (216*n^2 - 1098*n + 1407)*x + (270*n - 693)*x^2 + 54*x^3)*P[n - 1, x] + ((18*n - 51)*x - (54*n - 171)*x^2 - 54*x^3)*D[P[n - 1, x], x])/(6*n - 17 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(24*n - 44 + (216*n^2 - 1098*n + 1311)*x + (1944*n^3 - 15228*n^2 + 38934*n - 32481)*x^2 + (2916*n^2 - 16038*n + 22086)*x^3 + (972*n - 2592)*x^4)*v[i - 2] - ((18*n - 33)*x + (324*n^2 - 1566*n + 1800)*x^2 - (972*n^2 - 5022*n + 6318)*x^3 - (1458*n - 3807)*x^4 - 486*x^5)*deriv(v[i - 2], x) + (30*n - 79 + (216*n^2 - 1098*n + 1407)*x + (270*n - 693)*x^2 + 54*x^3)*v[i - 1] + ((18*n - 51)*x - (54*n - 171)*x^2 - 54*x^3)*deriv(v[i - 1], x))/(6*n - 17 + 6*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.