A391533 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(4 + 2\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 25 + 20\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	25	20	4								49
1	100	130	56	8							294
2	400	720	484	144	16						1764
3	1600	3680	3376	1544	352	32					10584
4	6400	17920	20864	12928	4496	832	64				63504
5	25600	84480	119296	93440	43840	12320	1920	128			381024
6	102400	389120	646144	612352	362240	136960	32320	4352	256		2286144
7	409600	1761280	3362816	3741696	2673664	1272320	403200	82048	9728	512	13716864

All 15 rows

Technical details

Node id: n-5c7e3a693e398f33 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-ac0f9723715f7eab

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      25,       20,        4;
  1:     100,      130,       56,         8;
  2:     400,      720,      484,       144,        16;
  3:    1600,     3680,     3376,      1544,       352,       32;
  4:    6400,    17920,    20864,     12928,      4496,      832,       64;
  5:   25600,    84480,   119296,     93440,     43840,    12320,     1920,      128;
  6:  102400,   389120,   646144,    612352,    362240,   136960,    32320,     4352,     256;
  7:  409600,  1761280,  3362816,   3741696,   2673664,  1272320,   403200,    82048,    9728,    512;
  8: 1638400,  7864320, 16973824,  21692416,  18178048, 10436608,  4157440,  1134592,  203008,  21504,  1024;
  9: 6553600, 34734080, 83623936, 120717312, 116097024, 78102528, 37502976, 12853248, 3081216, 492032, 47104, 2048;

%F Formula

The row polynomials satisfy P(n,x) = (4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 20*x + 4*x^2.

%t Mathematica

P[0, x_] := 25 + 20*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 2*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] = 25 + 20*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 2*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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    ↑ Liu–Wang theorem (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
  - ↑ Geometric basis map (elementary) (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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.