A200536 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 - 5\,x + 6\,x^{2} + 7\,n + 12\,n x\right)\,P_{n}(x) = \left(12 + 84\,x + 219\,x^{2} + 252\,x^{3} + 108\,x^{4} + 42\,n + 219\,n x + 378\,n x^{2} + 216\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	27	57	18													102
2	288	1566	2925	2052	324											7155
3	2484	22032	78489	141453	131706	55404	5832									437400
4	19440	239760	1267488	3729564	6619401	7140312	4426488	1329696	104976							24877125
5	143856	2274480	15969960	65396160	172288215	303367761	358237890	275649480	128070720	29918160	1889568					1353206250
6	1026432	19805472	173525328	911483280	3190570560	7819412922	13704801237	17207849628	15220863900	9101419200	3381381936	646232256	34012224			71412384375
7	7138368	162642816	1708484400	10957079952	47875468284	150536948688	350581136661	612644517765	804148978122	784404317844	554577342984	271159621488	83925162720	13570877376	612220032	3686871937500

All 15 rows

Technical details

Node id: n-41aa178d33268925 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-19ef812d44a90303

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:        27,         57,           18;
  2:       288,       1566,         2925,          2052,           324;
  3:      2484,      22032,        78489,        141453,        131706,          55404,            5832;
  4:     19440,     239760,      1267488,       3729564,       6619401,        7140312,         4426488,         1329696,          104976;
  5:    143856,    2274480,     15969960,      65396160,     172288215,      303367761,       358237890,       275649480,       128070720,         29918160,          1889568;
  6:   1026432,   19805472,    173525328,     911483280,    3190570560,     7819412922,     13704801237,     17207849628,     15220863900,       9101419200,       3381381936,        646232256,         34012224;
  7:   7138368,  162642816,   1708484400,   10957079952,   47875468284,   150536948688,    350581136661,    612644517765,    804148978122,     784404317844,     554577342984,     271159621488,      83925162720,      13570877376,       612220032;
  8:  48708864, 1279867392,  15677535744,  118766487168,  622342447776,  2389698648288,   6947669018304,  15577633391544,  27168173753553,   36879126013296,   38684387787696,   30847397894208,   18162714683232,    7511123499264,   1992163984128,    279172334592,    11019960576;
  9: 327525120, 9750170880, 136457042688, 1192483689984, 7288247064096, 33063299956512, 115296183526608, 315714478512384, 687409598738835, 1197202755113001, 1668482106359922, 1851259273447248, 1616916470097600, 1090276788692448, 549970417476288, 197047915059456, 45721816429824, 5653239775488, 198359290368;

%F Formula

The row polynomials satisfy (7*n-5+(12*n-5)*x+6*x^2)*P(n,x) = (42*n+12+(219*n+84)*x+(378*n+219)*x^2+(216*n+252)*x^3+108*x^4)*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[((42*n + 12 + (219*n + 84)*x + (378*n + 219)*x^2 + (216*n + 252)*x^3 + 108*x^4)*P[n - 1, x])/(7*n - 5 + (12*n - 5)*x + 6*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; for(n = 1, N, my(i = n + 1); v[i] = ((42*n + 12 + (219*n + 84)*x + (378*n + 219)*x^2 + (216*n + 252)*x^3 + 108*x^4)*v[i - 1])/(7*n - 5 + (12*n - 5)*x + 6*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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.