A128813 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(2 + 2\,x\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + n + 4\,n x + 3\,n x^{2} + n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	8	36	40													84
3	56	404	964	760												2184
4	616	6180	23128	38244	23560											91728
5	9856	127216	654328	1675792	2136184	1083760										5687136
6	216832	3429536	22537040	78744416	154246736	160558496	69360640									489093696
7	6288128	117887264	945084720	4199236464	11166430704	17767168944	15658930720	5895654400								55756681344
8	232660736	5047234720	47817846416	258385983648	870874710624	1874526197664	2516001851536	1924962661280	642626329600							8140475476224
9	10702393856	263814657216	2886044857056	18388982360384	75200730464832	204667165737408	370671648052960	430724534227776	291355733095680	87397180825600						1481566536672768
10	599334055936	16550218184192	205411745092992	1508866458452800	7263811977854336	23944682538456960	54732361803375488	85652067493546816	87816193735168896	53259293820116480	14507932017049600					328907771141354496
11	40155381747712	1228132095472128	17056080339884672	141970989989843008	786939827748347712	3049792313669629184	8432060065979092736	16630428520939358784	22928446411472132416	21043795239246414464	11570630915345502720	2887078471392870400				87489467123600295936
12	3172275158069248	106458950253010432	1636041389286839168	15223887089070495552	95531429039732576128	425864452300762417856	1382833938924711184384	3295337968659296136896	5719497968927047775104	7050644730596417860416	5859371723535202113920	2947177464346229900800	678463440777324544000			27471692676810492923904
13	291849314542370816	10663426816587933696	179685560183714061824	1848872952859079522816	12960236534060712785024	65355141168556868301824	243907582311482331455872	682067592382026109115648	1429116416553935536819072	2215801758700881533536768	2470938854748657088234624	1876608178968498530087680	869945261782380850867200	185898982772986925056000		10054639519712640410148864
14	30936027341491306496	1222547625953710149632	22416312253515477601280	252761170021116072954880	1958028925713904684422400	11023079708630213280060928	46506428334281097517698816	149373960802923184306314240	367019699347437417383366400	686475774053337072189724160	962112874352836215950488832	979737145071236484071435264	685222382302977905699630080	294607994897168962929971200	58744078556263868317696000	4243057877318734253082820608

Technical details

Node id: n-bee118c2057fa6b7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-4aae017df98b90e3

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:           1;
  1:           2,            4;
  2:           8,           36,            40;
  3:          56,          404,           964,            760;
  4:         616,         6180,         23128,          38244,          23560;
  5:        9856,       127216,        654328,        1675792,        2136184,         1083760;
  6:      216832,      3429536,      22537040,       78744416,      154246736,       160558496,        69360640;
  7:     6288128,    117887264,     945084720,     4199236464,    11166430704,     17767168944,     15658930720,      5895654400;
  8:   232660736,   5047234720,   47817846416,   258385983648,   870874710624,   1874526197664,   2516001851536,   1924962661280,    642626329600;
  9: 10702393856, 263814657216, 2886044857056, 18388982360384, 75200730464832, 204667165737408, 370671648052960, 430724534227776, 291355733095680, 87397180825600;

%F Formula

The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(4*n^2+4*n+4)*x+(3*n^2+3*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(2 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*v[i - 1])/(2 + 2*x)); 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, 0]
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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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.