A038328 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 70\,x + 16\,x^{2} + 42\,n x + 16\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 1764\,x + 952\,x^{2} + 144\,x^{3} + 588\,n x + 504\,n x^{2} + 96\,n x^{3} + 56\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 448\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	196	448	144								788
1	2744	9408	6048	1024							19224
2	38416	175616	169344	57344	6400						447120
3	537824	3073280	3951360	2007040	448000	36864					10054368
4	7529536	51631104	82978560	56197120	18816000	3096576	200704				220449600
5	105413504	843308032	1626379776	1376829440	614656000	151732224	19668992	1048576			4739036544
6	1475789056	13492928512	30359089152	30840979456	17210368000	5664669696	1101463552	117440512	5308416		100268036352
7	20661046784	212513624064	546463604736	647660568576	433701273600	178437095424	46261469184	7398752256	668860416	26214400	2093792509440

All 15 rows

Technical details

Node id: n-906b14404b065035 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-2b0c83ee9d9e3a42

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:           196,            448,             144;
  1:          2744,           9408,            6048,            1024;
  2:         38416,         175616,          169344,           57344,            6400;
  3:        537824,        3073280,         3951360,         2007040,          448000,           36864;
  4:       7529536,       51631104,        82978560,        56197120,        18816000,         3096576,         200704;
  5:     105413504,      843308032,      1626379776,      1376829440,       614656000,       151732224,       19668992,        1048576;
  6:    1475789056,    13492928512,     30359089152,     30840979456,     17210368000,      5664669696,     1101463552,      117440512,       5308416;
  7:   20661046784,   212513624064,    546463604736,    647660568576,    433701273600,    178437095424,    46261469184,     7398752256,     668860416,     26214400;
  8:  289254654976,  3305767485440,   9563113082880,  12953211371520,  10119696384000,   4996238671872,  1619151421440,   345275105280,   46820229120,   3670016000,   126877696;
  9: 4049565169664, 50908819275776, 163635490529280, 249349318901760, 222633320448000, 128236792578048, 49869863780352, 13293091553280, 2403438428160, 282591232000, 19539165184, 603979776;

%F Formula

The row polynomials satisfy (49+(42*n+70)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = (686+(588*n+1764)*x+(56*n^2+504*n+952)*x^2+(16*n^2+96*n+144)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 448*x + 144*x^2.

%t Mathematica

P[0, x_] := 196 + 448*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (588*n + 1764)*x + (56*n^2 + 504*n + 952)*x^2 + (16*n^2 + 96*n + 144)*x^3)*P[n - 1, x])/(49 + (42*n + 70)*x + (4*n^2 + 16*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 448*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (588*n + 1764)*x + (56*n^2 + 504*n + 952)*x^2 + (16*n^2 + 96*n + 144)*x^3)*v[i - 1])/(49 + (42*n + 70)*x + (4*n^2 + 16*n + 16)*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
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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

Related ?

Same fingerprint ?

A038255 → basis A013609 → (θ+1)²

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.