A241171 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 9\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - 2\,x^{4}\right)\,P_{n-2}''(x) + \left(1 + 12\,x\right)\,P_{n-1}(x) + \left(3\,x + 11\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 1 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	12														13
2	1	60	270													331
3	1	252	3780	10080												14113
4	1	1020	39690	302400	567000											910111
5	1	4092	380160	6320160	31185000	44906400										82795813
6	1	16380	3513510	115315200	1135134000	4086482400	4767562800									10108024291
7	1	65532	31990140	1976214240	34864830000	237015979200	667458792000	653837184000								1595185055113
8	1	262140	289385730	32813726400	982782801000	11293869787200	57868677266400	133382785536000	112541725296000							316102944060871
9	1	1048572	2610369720	535823356320	26415342135000	484415710178400	4034597695128000	16472774013696000	32074391709360000	23758808673600000						76851941578872013
10	1	4194300	23516920350	8670627504000	690523617168000	19531060663514400	249124088291878800	1607573792208384000	5433401955565584000	9147141339336000000	6037113283961760000					22494584737792907851
11	1	16777212	211746654900	139608005090400	17750813226300000	757807654366224000	14280627933411840000	137050311952794240000	716809197384680400000	2061765657886334400000	3054779321684650560000	1817719926952849920000				7803180742082082406513
12	1	67108860	1906097381370	2241633289896000	451623280943835000	28686939964707024000	780204681375727608000	10729706081561026560000	81651329933033820960000	357502951903177080000000	893522951592760288800000	1181517952519352448000000	639988890947982576000000			3165723128466028073173231
13	1	268435452	17156386381680	35937293607240480	11416673896152525000	1068498402580204596000	41275626180330381840000	793700688380005486080000	8460608383864431194400000	52987798176180403536000000	198957777221321290972800000	439170222951443304921600000	524150901686397729744000000	260524708530517214784000000		1485088073215315637639898613
14	1	1073741820	154413517232790	575637202807430400	287438320169220402000	39370143065462645436000	2135944596576787098102000	56457501361051374904320000	821925010080427371037920000	7084908254211642494640000000	37538090560347316746206400000	123144890119282028245248000000	243206018382488546601216000000	264432579158474973005760000000	121423123082973201890400000000	797710167671972941765774985011

Technical details

Node id: n-3c9286c555c35626 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-403c190250d8ebb3

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: 1,      12;
  2: 1,      60,        270;
  3: 1,     252,       3780,        10080;
  4: 1,    1020,      39690,       302400,         567000;
  5: 1,    4092,     380160,      6320160,       31185000,        44906400;
  6: 1,   16380,    3513510,    115315200,     1135134000,      4086482400,       4767562800;
  7: 1,   65532,   31990140,   1976214240,    34864830000,    237015979200,     667458792000,      653837184000;
  8: 1,  262140,  289385730,  32813726400,   982782801000,  11293869787200,   57868677266400,   133382785536000,   112541725296000;
  9: 1, 1048572, 2610369720, 535823356320, 26415342135000, 484415710178400, 4034597695128000, 16472774013696000, 32074391709360000, 23758808673600000;

%F Formula

The row polynomials satisfy P(n,x) = -6*x^2*P(n-2,x) - (2*x^2+9*x^3)*d/dx P(n-2,x) - (x^3+2*x^4)*d^2/dx^2 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x+11*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[-6*x^2*P[n - 2, x] - (2*x^2 + 9*x^3)*D[P[n - 2, x], x] - (x^3 + 2*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 12*x)*P[n - 1, x] + (3*x + 11*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
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] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -6*x^2*v[i - 2] - (2*x^2 + 9*x^3)*deriv(v[i - 2], x) - (x^3 + 2*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 12*x)*v[i - 1] + (3*x + 11*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.