A159854 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 30\,x + 192\,x^{2} + 384\,x^{3} + 256\,x^{4} + 6\,n x + 80\,n x^{2} + 288\,n x^{3} + 256\,n x^{4} + 4\,n^{2} x^{2} + 32\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n}(x) = \left(2 + 76\,x + 696\,x^{2} + 2512\,x^{3} + 3968\,x^{4} + 2304\,x^{5} + 12\,n x + 200\,n x^{2} + 1056\,n x^{3} + 2176\,n x^{4} + 1536\,n x^{5} + 8\,n^{2} x^{2} + 80\,n^{2} x^{3} + 256\,n^{2} x^{4} + 256\,n^{2} x^{5}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 32\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	32	144								177
1	2	80	576	1024							1682
2	4	192	1872	6144	6400						14612
3	8	448	5472	25600	51200	36864					119592
4	16	1024	14976	90112	262400	368640	200704				937872
5	32	2304	39168	286720	1088000	2248704	2408448	1048576			7121952
6	64	5120	99072	851968	3968000	10764288	17059840	14680064	5308416		52736832
7	128	11264	244224	2408448	13260800	44384256	92725248	118489088	84934656	26214400	382672512

All 15 rows

Technical details

Node id: n-f9c72d93728b9394 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-225246347a30c19a

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,    32,     144;
  1:   2,    80,     576,     1024;
  2:   4,   192,    1872,     6144,      6400;
  3:   8,   448,    5472,    25600,     51200,     36864;
  4:  16,  1024,   14976,    90112,    262400,    368640,     200704;
  5:  32,  2304,   39168,   286720,   1088000,   2248704,    2408448,    1048576;
  6:  64,  5120,   99072,   851968,   3968000,  10764288,   17059840,   14680064,    5308416;
  7: 128, 11264,  244224,  2408448,  13260800,  44384256,   92725248,  118489088,   84934656,   26214400;
  8: 256, 24576,  589824,  6553600,  41574400, 165150720,  427098112,  721420288,  769720320,  471859200,  126877696;
  9: 512, 53248, 1400832, 17301504, 124108800, 569769984, 1753350144, 3674210304, 5191630848, 4744806400, 2537553920, 603979776;

%F Formula

The row polynomials satisfy (1+(6*n+30)*x+(4*n^2+80*n+192)*x^2+(32*n^2+288*n+384)*x^3+(64*n^2+256*n+256)*x^4)*P(n,x) = (2+(12*n+76)*x+(8*n^2+200*n+696)*x^2+(80*n^2+1056*n+2512)*x^3+(256*n^2+2176*n+3968)*x^4+(256*n^2+1536*n+2304)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 32*x + 144*x^2.

%t Mathematica

P[0, x_] := 1 + 32*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (12*n + 76)*x + (8*n^2 + 200*n + 696)*x^2 + (80*n^2 + 1056*n + 2512)*x^3 + (256*n^2 + 2176*n + 3968)*x^4 + (256*n^2 + 1536*n + 2304)*x^5)*P[n - 1, x])/(1 + (6*n + 30)*x + (4*n^2 + 80*n + 192)*x^2 + (32*n^2 + 288*n + 384)*x^3 + (64*n^2 + 256*n + 256)*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 + 32*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (12*n + 76)*x + (8*n^2 + 200*n + 696)*x^2 + (80*n^2 + 1056*n + 2512)*x^3 + (256*n^2 + 2176*n + 3968)*x^4 + (256*n^2 + 1536*n + 2304)*x^5)*v[i - 1])/(1 + (6*n + 30)*x + (4*n^2 + 80*n + 192)*x^2 + (32*n^2 + 288*n + 384)*x^3 + (64*n^2 + 256*n + 256)*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_A013611 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.