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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	289254654976	3305767485440	9563113082880	12953211371520	10119696384000	4996238671872	1619151421440	345275105280	46820229120	3670016000	126877696							43242325300224
9	4049565169664	50908819275776	163635490529280	249349318901760	222633320448000	128236792578048	49869863780352	13293091553280	2403438428160	282591232000	19539165184	603979776						884682435041280
10	56693912375296	777516512575488	2749076240891904	4654520619499520	4675299729408000	3077683021873152	1396356185849856	446647876190208	100944413982720	15825108992000	1641289875456	101468602368	2835349504					17952309215465472
11	793714773254144	11792333774061568	45484715985666048	84712275274891264	94544950083584000	70017288747614208	36305260832096256	13548318911102976	3674376668971008	720042459136000	99571585777664	9233642815488	516033609728	13153337344				361702611925917696
12	11112006825558016	177792109208928256	742917027765878784	1509418723079880704	1853081021638246400	1524820954948042752	889478890386358272	379352929510883328	120029637853052928	28225664398131200	4879007703105536	603264663945216	50571293753344	2578054119424	60397977600			7241764447727861760
13	155568095557812224	2666881638133923840	12000967371602657280	26414827653897912320	35377001322184704000	32021240053908897792	20754507442348359680	9958014399660687360	3600889135591587840	987898253934592000	204918323530432512	31671394857123840	3539990562734080	270695682539520	12683575296000	274877906944		144178208729907167232
14	2177953337809371136	39825432462799929344	192015477945642516480	455147799574856335360	660370691347447808000	652068888370508464128	464900966708603256832	247843913947110440960	100824895796564459520	31612744125906944000	7650284078469480448	1418878489599148032	198239471513108480	20211944296284160	1420560433152000	61572651155456	1241245548544	2856077860975457402880

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.