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

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 30\,x - 9\,x^{2} + 30\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 6\,x\)

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	6														7
2	4	30	27													61
3	28	234	324	108												694
4	280	2508	4293	2376	405											9862
5	3640	34284	67095	48060	14175	1458										168712
6	58240	570384	1227798	1037340	407025	74358	5103									3380248
7	1106560	11186736	25894890	24620652	11623500	2878092	357210	17496								77685136
8	24344320	252747552	620027892	645233904	348044445	105162624	17931942	1609632	59049							2015161360
9	608608000	6464754720	16638061284	18610959168	11120738265	3882025602	816367734	101721744	6908733	196830						58250342080
10	17041024000	184664780160	494957112192	587659101840	381171768300	148731374610	36445386159	5647183920	536755410	28540350	649539					1856883676480
11	528271744000	5826854328960	16174661988672	20197260605808	14020046449200	5982890978790	1650366782064	300018311208	35698663440	2673935550	114318864	2125764				64718860232320
12	17961239296000	201282677648640	576159352095168	751405508552160	552421306544580	253890460495980	77052589015941	15859022976720	2226316357287	209909353500	12710828691	446410440	6908733			2448481546483840
13	664565853952000	7555226508775680	22223668076940096	30106641224810592	23257358999220060	11382663741911748	3739562405325747	850964155336152	135897907766049	15187700603790	1163001528117	58116261996	1706457051	22320522		99932959421209600
14	26582634158080000	306196455474739200	922945242367094400	1293160321300184064	1043194264561842120	539033042073662136	189421819309720998	46859923031705784	8307920334901473	1060501050038430	96639473117187	6130837299132	257136133527	6405989814	71744535	4376865157586252800

Technical details

Node id: n-7a174e012bdbab42 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5e4d568b2cdb4901

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,          6;
  2:         4,         30,          27;
  3:        28,        234,         324,         108;
  4:       280,       2508,        4293,        2376,         405;
  5:      3640,      34284,       67095,       48060,       14175,       1458;
  6:     58240,     570384,     1227798,     1037340,      407025,      74358,      5103;
  7:   1106560,   11186736,    25894890,    24620652,    11623500,    2878092,    357210,     17496;
  8:  24344320,  252747552,   620027892,   645233904,   348044445,  105162624,  17931942,   1609632,   59049;
  9: 608608000, 6464754720, 16638061284, 18610959168, 11120738265, 3882025602, 816367734, 101721744, 6908733, 196830;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(18*n-30)*x+9*x^2)*P(n-2,x) + (6*n-7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*P[n - 2, x] + (6*n - 7 + 6*x)*P[n - 1, 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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (18*n - 30)*x + 9*x^2)*v[i - 2] + (6*n - 7 + 6*x)*v[i - 1]); 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.