A272867 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 2\,x^{2} + 6\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(18 + 72\,x + 96\,x^{2} + 48\,x^{3} + 8\,x^{4} + 36\,n x + 96\,n x^{2} + 72\,n x^{3} + 16\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	24	12													42
2	36	288	576	384	80											1364
3	216	2592	9072	13824	10080	3456	448									39688
4	1296	20736	103680	248832	328320	248832	107520	24576	2304							1086096
5	7776	155520	1010880	3317760	6393600	7713792	5967360	2949120	898560	153600	11264					28579232
6	46656	1119744	8957952	37324800	94867200	158257152	179601408	140673024	75893760	27648000	6488064	884736	53248			731815744
7	279936	7838208	74462976	376233984	1188794880	2539579392	3824464896	4151181312	3278112768	1881169920	774918144	222953472	42491904	4816896	245760	18367544448

All 15 rows

Technical details

Node id: n-a9f53fa0d0ff53f6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4b61633885bd0ded

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:        6,        24,         12;
  2:       36,       288,        576,         384,           80;
  3:      216,      2592,       9072,       13824,        10080,         3456,          448;
  4:     1296,     20736,     103680,      248832,       328320,       248832,       107520,         24576,          2304;
  5:     7776,    155520,    1010880,     3317760,      6393600,      7713792,      5967360,       2949120,        898560,        153600,         11264;
  6:    46656,   1119744,    8957952,    37324800,     94867200,    158257152,    179601408,     140673024,      75893760,      27648000,       6488064,        884736,        53248;
  7:   279936,   7838208,   74462976,   376233984,   1188794880,   2539579392,   3824464896,    4151181312,    3278112768,    1881169920,     774918144,     222953472,     42491904,      4816896,      245760;
  8:  1679616,  53747712,  591224832,  3511517184,  13272698880,  34613526528,  65158152192,   90901315584,   95378300928,   75751096320,   45507280896,   20511719424,   6816595968,   1618477056,   259522560,    25165824,    1114112;
  9: 10077696, 362797056, 4534963200, 30958682112, 136250449920, 419877126144, 949865398272, 1623610884096, 2134536634368, 2180723834880, 1739252072448, 1082407256064, 522677256192, 193523613696, 53827338240, 10871635968, 1504051200, 127401984, 4980736;

%F Formula

The row polynomials satisfy (3+6*n*x+(4*n-2)*x^2)*P(n,x) = (18+(36*n+72)*x+(96*n+96)*x^2+(72*n+48)*x^3+(16*n+8)*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[((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*P[n - 1, x])/(3 + 6*n*x + (4*n - 2)*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] = ((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*v[i - 1])/(3 + 6*n*x + (4*n - 2)*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 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 basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.