basis-map coefficients of A297191, shifted → basis A008288 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 17\,x - 35\,x^{2} - 11\,x^{3} + 3\,x^{4} - 8\,n + 88\,n x + 120\,n x^{2} - 24\,n x^{3} - 16\,n x^{4} + 4\,n^{2} - 120\,n^{2} x - 32\,n^{2} x^{2} + 152\,n^{2} x^{3} + 28\,n^{2} x^{4} + 48\,n^{3} x - 112\,n^{3} x^{2} - 176\,n^{3} x^{3} - 16\,n^{3} x^{4} + 64\,n^{4} x^{2} + 64\,n^{4} x^{3}\right)\,P_{n}(x) = \left(x^{3} - 5\,x^{4} - 5\,x^{5} + x^{6} + 8\,n x^{3} + 24\,n x^{4} - 8\,n x^{5} + 8\,n x^{6} - 4\,n^{2} x^{2} - 24\,n^{2} x^{3} - 16\,n^{2} x^{4} - 8\,n^{2} x^{5} + 20\,n^{2} x^{6} - 48\,n^{3} x^{3} - 144\,n^{3} x^{4} - 80\,n^{3} x^{5} + 16\,n^{3} x^{6} - 64\,n^{4} x^{4} - 64\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(4 + 47\,x + 253\,x^{2} + 504\,x^{3} + 276\,x^{4} + 49\,x^{5} + 3\,x^{6} - 8\,n - 40\,n x - 40\,n x^{2} + 240\,n x^{3} + 360\,n x^{4} + 120\,n x^{5} + 8\,n x^{6} + 4\,n^{2} - 56\,n^{2} x - 492\,n^{2} x^{2} - 1072\,n^{2} x^{3} - 660\,n^{2} x^{4} - 88\,n^{2} x^{5} - 4\,n^{2} x^{6} + 48\,n^{3} x + 208\,n^{3} x^{2} + 64\,n^{3} x^{3} - 192\,n^{3} x^{4} - 112\,n^{3} x^{5} - 16\,n^{3} x^{6} + 64\,n^{4} x^{2} + 320\,n^{4} x^{3} + 320\,n^{4} x^{4} + 64\,n^{4} x^{5}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 12\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	12	9																											22
2	1	28	117	112	25																									283
3	1	44	369	1008	1025	396	49																							2892
4	1	60	765	3696	8025	8316	4165	960	81																					26069
5	1	76	1305	9200	32225	60588	63161	36800	11745	1900	121																			217122
6	1	92	1989	18544	91225	258588	440461	459712	295569	115900	26741	3312	169																	1712303
7	1	108	2817	32752	209025	805068	1963969	3112896	3246561	2236300	1011681	294768	52897	5292	225															12974360
8	1	124	3789	52848	416025	2041020	6578005	14345152	21524049	22414300	16243645	8164080	2812329	647388	94725	7936	289													95345705
9	1	140	4905	79856	749025	4482540	18095945	50909120	101773665	146256300	152032265	114545520	62412545	24404940	6741225	1277696	157505	11340	361											683934894
10	1	156	6165	114800	1251225	8857692	43153565	149791680	378540945	705973500	979791813	1016601840	789795305	458737020	198154125	62988032	14464161	2324700	247285	15600	441									4810810051
11	1	172	7569	158704	1972225	16143372	92363089	383025088	1178283105	2729815500	4810146033	6486727536	6718303137	5350438380	3273008625	1532100352	544753729	145290348	28478929	3967600	370881	20812	529							33295375716
12	1	188	9117	212592	2968025	27602172	181667941	878527424	3198693969	8912924700	19217129613	32298945264	42519718761	43962453276	35734331925	22817087232	11412624161	4447545084	1338410749	306690800	52355961	6430908	535877	27072	625					227316893437
13	1	204	10809	277488	4301025	44819244	333898201	1848020928	7793865441	25483166700	65373829785	132682512240	214267473537	276346989324	285267346425	235880021760	156140800065	82565460108	34735622521	11550130800	3005083809	602569836	91009689	9989568	750625	34476	729			1534027985338
14	1	220	12645	354416	6040025	69739164	580526765	3623076800	17401117329	65486224700	195576748565	467744701680	901604952105	1406978398140	1782865856925	1837686071040	1541797817505	1052425578780	583497572165	261944898800	94739416569	27399518300	6267319565	1115826624	151000625	14974076	1024245	43120	841	10248978811735

Technical details

Node id: n-2124eaaf0381b900 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f17030395a2c695a

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  12,    9;
  2: 1,  28,  117,   112,     25;
  3: 1,  44,  369,  1008,   1025,     396,       49;
  4: 1,  60,  765,  3696,   8025,    8316,     4165,      960,        81;
  5: 1,  76, 1305,  9200,  32225,   60588,    63161,    36800,     11745,      1900,       121;
  6: 1,  92, 1989, 18544,  91225,  258588,   440461,   459712,    295569,    115900,     26741,      3312,      169;
  7: 1, 108, 2817, 32752, 209025,  805068,  1963969,  3112896,   3246561,   2236300,   1011681,    294768,    52897,     5292,     225;
  8: 1, 124, 3789, 52848, 416025, 2041020,  6578005, 14345152,  21524049,  22414300,  16243645,   8164080,  2812329,   647388,   94725,    7936,    289;
  9: 1, 140, 4905, 79856, 749025, 4482540, 18095945, 50909120, 101773665, 146256300, 152032265, 114545520, 62412545, 24404940, 6741225, 1277696, 157505, 11340, 361;

%F Formula

The row polynomials satisfy (4*n^2-8*n+4+(48*n^3-120*n^2+88*n-17)*x+(64*n^4-112*n^3-32*n^2+120*n-35)*x^2+(64*n^4-176*n^3+152*n^2-24*n-11)*x^3-(16*n^3-28*n^2+16*n-3)*x^4)*P(n,x) = -(4*n^2*x^2+(48*n^3+24*n^2-8*n-1)*x^3+(64*n^4+144*n^3+16*n^2-24*n+5)*x^4+(64*n^4+80*n^3+8*n^2+8*n+5)*x^5-(16*n^3+20*n^2+8*n+1)*x^6)*P(n-2,x) + (4*n^2-8*n+4+(48*n^3-56*n^2-40*n+47)*x+(64*n^4+208*n^3-492*n^2-40*n+253)*x^2+(320*n^4+64*n^3-1072*n^2+240*n+504)*x^3+(320*n^4-192*n^3-660*n^2+360*n+276)*x^4+(64*n^4-112*n^3-88*n^2+120*n+49)*x^5-(16*n^3+4*n^2-8*n-3)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 12*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*P[n - 2, x] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*P[n - 1, x])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*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; v[2] = 1 + 12*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2*x^2 + (48*n^3 + 24*n^2 - 8*n - 1)*x^3 + (64*n^4 + 144*n^3 + 16*n^2 - 24*n + 5)*x^4 + (64*n^4 + 80*n^3 + 8*n^2 + 8*n + 5)*x^5 - (16*n^3 + 20*n^2 + 8*n + 1)*x^6)*v[i - 2] + (4*n^2 - 8*n + 4 + (48*n^3 - 56*n^2 - 40*n + 47)*x + (64*n^4 + 208*n^3 - 492*n^2 - 40*n + 253)*x^2 + (320*n^4 + 64*n^3 - 1072*n^2 + 240*n + 504)*x^3 + (320*n^4 - 192*n^3 - 660*n^2 + 360*n + 276)*x^4 + (64*n^4 - 112*n^3 - 88*n^2 + 120*n + 49)*x^5 - (16*n^3 + 4*n^2 - 8*n - 3)*x^6)*v[i - 1])/(4*n^2 - 8*n + 4 + (48*n^3 - 120*n^2 + 88*n - 17)*x + (64*n^4 - 112*n^3 - 32*n^2 + 120*n - 35)*x^2 + (64*n^4 - 176*n^3 + 152*n^2 - 24*n - 11)*x^3 - (16*n^3 - 28*n^2 + 16*n - 3)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    no recorded property covers this requirement
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.?
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)
    
    ↑ Affine-current basis map theorem (above)

References

[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.