basis-map coefficients of A371080, shifted → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 60\,x + 81\,x^{2} + 30\,n - 36\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-60\,x^{2} + 36\,x^{3} + 36\,n x^{2}\right)\,P_{n-2}'(x) + \left(-7 + 12\,x + 6\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	4	60	81													145
3	28	468	972	432												1900
4	280	5016	12879	9504	2025											29704
5	3640	68568	201285	192240	70875	8748										545356
6	58240	1140768	3683394	4149360	2035125	446148	35721									11548756
7	1106560	22373472	77684670	98482608	58117500	17268552	2500470	139968								277673800
8	24344320	505495104	1860083676	2580935616	1740222225	630975744	125523594	12877056	531441							7480988776
9	608608000	12929509440	49914183852	74443836672	55603691325	23292153612	5714574138	813773952	62178597	1968300						223384477888
10	17041024000	369329560320	1484871336576	2350636407360	1905858841500	892388247660	255117703113	45177471360	4830798690	285403500	7144929					7325543939008
11	528271744000	11653708657920	48523985966016	80789042423232	70100232246000	35897345872740	11552567474448	2400146489664	321287970960	26739355500	1257507504	25509168				261794611217152
12	17961239296000	402565355297280	1728478056285504	3005622034208640	2762106532722900	1523342762975880	539368123111587	126872183813760	20036847215583	2099093535000	139819115601	5356925280	89813529			10128597494316544
13	664565853952000	15110453017551360	66671004230820288	120426564899242368	116286794996100300	68295982451470488	26176936837280229	6807713242689216	1223081169894441	151877006037900	12793016809287	697395143952	22183941663	312487308		421828486613420800
14	26582634158080000	612392910949478400	2768835727101283200	5172641285200736256	5215971322809210600	3234198252441972816	1325952735168046986	374879384253646272	74771283014113257	10605010500384300	1063034204289057	73570047589584	3342769735851	89683857396	1076168025	18817970583378592000

Technical details

Node id: n-5fed1c787247e740 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3ba7e13d879efe8a

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:         4,          60,          81;
  3:        28,         468,         972,         432;
  4:       280,        5016,       12879,        9504,        2025;
  5:      3640,       68568,      201285,      192240,       70875,        8748;
  6:     58240,     1140768,     3683394,     4149360,     2035125,      446148,      35721;
  7:   1106560,    22373472,    77684670,    98482608,    58117500,    17268552,    2500470,    139968;
  8:  24344320,   505495104,  1860083676,  2580935616,  1740222225,   630975744,  125523594,  12877056,   531441;
  9: 608608000, 12929509440, 49914183852, 74443836672, 55603691325, 23292153612, 5714574138, 813773952, 62178597, 1968300;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(36*n-60)*x-81*x^2)*P(n-2,x) + ((36*n-60)*x^2+36*x^3)*d/dx P(n-2,x) + (6*n-7+12*x)*P(n-1,x) - 12*x^2*d/dx 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[-(9*n^2 - 30*n + 25 + (36*n - 60)*x - 81*x^2)*P[n - 2, x] + ((36*n - 60)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (6*n - 7 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], 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] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (36*n - 60)*x - 81*x^2)*v[i - 2] + ((36*n - 60)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (6*n - 7 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 basis_map_A013610 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[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.