A159854 → 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

\(\left(4 + 90\,x + 432\,x^{2} + 648\,x^{3} + 324\,x^{4} + 18\,n x + 180\,n x^{2} + 486\,n x^{3} + 324\,n x^{4} + 9\,n^{2} x^{2} + 54\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n}(x) = \left(8 + 228\,x + 1566\,x^{2} + 4239\,x^{3} + 5022\,x^{4} + 2187\,x^{5} + 36\,n x + 450\,n x^{2} + 1782\,n x^{3} + 2754\,n x^{4} + 1458\,n x^{5} + 18\,n^{2} x^{2} + 135\,n^{2} x^{3} + 324\,n^{2} x^{4} + 243\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 24\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	24	81								106
1	2	60	324	432							818
2	4	144	1053	2592	2025						5818
3	8	336	3078	10800	16200	8748					39170
4	16	768	8424	38016	83025	87480	35721				253450
5	32	1728	22032	120960	344250	533628	428652	139968			1591250
6	64	3840	55728	359424	1255500	2554416	3036285	1959552	531441		9756250
7	128	8448	137376	1016064	4195800	10532592	16503102	15816384	8503056	1968300	58681250

All 15 rows

Technical details

Node id: n-7ceebe2d3eabc45a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-540508e6fe278536

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1,    24,     81;
  1:   2,    60,    324,     432;
  2:   4,   144,   1053,    2592,     2025;
  3:   8,   336,   3078,   10800,    16200,      8748;
  4:  16,   768,   8424,   38016,    83025,     87480,     35721;
  5:  32,  1728,  22032,  120960,   344250,    533628,    428652,    139968;
  6:  64,  3840,  55728,  359424,  1255500,   2554416,   3036285,   1959552,    531441;
  7: 128,  8448, 137376, 1016064,  4195800,  10532592,  16503102,  15816384,   8503056,   1968300;
  8: 256, 18432, 331776, 2764800, 13154400,  39191040,  76014288,  96297984,  77058945,  35429400,   7144929;
  9: 512, 39936, 787968, 7299072, 39268800, 135209088, 312058656, 490447872, 519749298, 356262300, 142898580, 25509168;

%F Formula

The row polynomials satisfy (4+(18*n+90)*x+(9*n^2+180*n+432)*x^2+(54*n^2+486*n+648)*x^3+(81*n^2+324*n+324)*x^4)*P(n,x) = (8+(36*n+228)*x+(18*n^2+450*n+1566)*x^2+(135*n^2+1782*n+4239)*x^3+(324*n^2+2754*n+5022)*x^4+(243*n^2+1458*n+2187)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 81*x^2.

%t Mathematica

P[0, x_] := 1 + 24*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (36*n + 228)*x + (18*n^2 + 450*n + 1566)*x^2 + (135*n^2 + 1782*n + 4239)*x^3 + (324*n^2 + 2754*n + 5022)*x^4 + (243*n^2 + 1458*n + 2187)*x^5)*P[n - 1, x])/(4 + (18*n + 90)*x + (9*n^2 + 180*n + 432)*x^2 + (54*n^2 + 486*n + 648)*x^3 + (81*n^2 + 324*n + 324)*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 + 24*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (36*n + 228)*x + (18*n^2 + 450*n + 1566)*x^2 + (135*n^2 + 1782*n + 4239)*x^3 + (324*n^2 + 2754*n + 5022)*x^4 + (243*n^2 + 1458*n + 2187)*x^5)*v[i - 1])/(4 + (18*n + 90)*x + (9*n^2 + 180*n + 432)*x^2 + (54*n^2 + 486*n + 648)*x^3 + (81*n^2 + 324*n + 324)*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
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 Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

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.