A272867 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x - 6\,x^{2} - 4\,x^{3} + x^{4} + 12\,n x + 8\,n x^{2} - 4\,n x^{3} - 4\,n x^{4} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(1 + 12\,x + 51\,x^{2} + 88\,x^{3} + 51\,x^{4} + 12\,x^{5} + x^{6} + 12\,n x + 88\,n x^{2} + 200\,n x^{3} + 156\,n x^{4} + 44\,n x^{5} + 4\,n x^{6} + 16\,n^{2} x^{2} + 80\,n^{2} x^{3} + 84\,n^{2} x^{4} + 32\,n^{2} x^{5} + 4\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	16	9													26
2	1	32	162	128	25											348
3	1	48	459	1408	1275	432	49									3672
4	1	64	900	4864	11350	10944	4900	1024	81							34128
5	1	80	1485	11520	44250	87264	86730	46080	13365	2000	121					292896
6	1	96	2214	22400	120375	368064	648564	654336	390015	140000	29766	3456	169			2379456
7	1	112	3087	38528	266525	1111824	2888403	4697088	4774707	3088400	1289981	346752	57967	5488	225	18569088

All 15 rows

Technical details

Node id: n-4b769f167be1b9f8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d7a387249c78d2a7

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: 1,  16,    9;
  2: 1,  32,  162,   128,     25;
  3: 1,  48,  459,  1408,   1275,     432,       49;
  4: 1,  64,  900,  4864,  11350,   10944,     4900,     1024,        81;
  5: 1,  80, 1485, 11520,  44250,   87264,    86730,    46080,     13365,      2000,       121;
  6: 1,  96, 2214, 22400, 120375,  368064,   648564,   654336,    390015,    140000,     29766,      3456,      169;
  7: 1, 112, 3087, 38528, 266525, 1111824,  2888403,  4697088,   4774707,   3088400,   1289981,    346752,    57967,     5488,     225;
  8: 1, 128, 4104, 60928, 515900, 2733696,  9464056, 21764096,  33328422,  34006400,  23370424,  10934784,  3487484,   746368,  102600,    8192,    289;
  9: 1, 144, 5265, 90624, 908100, 5842368, 25358676, 76068864, 159153822, 232597600, 237748302, 171154944, 87461556, 31808448, 8172900, 1449984, 169065, 11664, 361;

%F Formula

The row polynomials satisfy (1+(12*n-4)*x+(16*n^2+8*n-6)*x^2+(16*n^2-4*n-4)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (1+(12*n+12)*x+(16*n^2+88*n+51)*x^2+(80*n^2+200*n+88)*x^3+(84*n^2+156*n+51)*x^4+(32*n^2+44*n+12)*x^5+(4*n^2+4*n+1)*x^6)*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[((1 + (12*n + 12)*x + (16*n^2 + 88*n + 51)*x^2 + (80*n^2 + 200*n + 88)*x^3 + (84*n^2 + 156*n + 51)*x^4 + (32*n^2 + 44*n + 12)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(1 + (12*n - 4)*x + (16*n^2 + 8*n - 6)*x^2 + (16*n^2 - 4*n - 4)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];
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] = ((1 + (12*n + 12)*x + (16*n^2 + 88*n + 51)*x^2 + (80*n^2 + 200*n + 88)*x^3 + (84*n^2 + 156*n + 51)*x^4 + (32*n^2 + 44*n + 12)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(1 + (12*n - 4)*x + (16*n^2 + 8*n - 6)*x^2 + (16*n^2 - 4*n - 4)*x^3 + (4*n^2 - 4*n + 1)*x^4)); 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, 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.