A272867 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x - 6\,x^{2} - 4\,x^{3} + x^{4} + 12\,n x + 8\,n x^{2} - 4\,n x^{3} - 4\,n x^{4} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(1 + 12\,x + 51\,x^{2} + 88\,x^{3} + 51\,x^{4} + 12\,x^{5} + x^{6} + 12\,n x + 88\,n x^{2} + 200\,n x^{3} + 156\,n x^{4} + 44\,n x^{5} + 4\,n x^{6} + 16\,n^{2} x^{2} + 80\,n^{2} x^{3} + 84\,n^{2} x^{4} + 32\,n^{2} x^{5} + 4\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	16	9																											26
2	1	32	162	128	25																									348
3	1	48	459	1408	1275	432	49																							3672
4	1	64	900	4864	11350	10944	4900	1024	81																					34128
5	1	80	1485	11520	44250	87264	86730	46080	13365	2000	121																			292896
6	1	96	2214	22400	120375	368064	648564	654336	390015	140000	29766	3456	169																	2379456
7	1	112	3087	38528	266525	1111824	2888403	4697088	4774707	3088400	1289981	346752	57967	5488	225															18569088
8	1	128	4104	60928	515900	2733696	9464056	21764096	33328422	34006400	23370424	10934784	3487484	746368	102600	8192	289													140527872
9	1	144	5265	90624	908100	5842368	25358676	76068864	159153822	232597600	237748302	171154944	87461556	31808448	8172900	1449984	169065	11664	361											1038002688
10	1	160	6570	128640	1489125	11276928	58947000	218941440	586171890	1137400000	1601268988	1637856000	1223000610	670508160	270675000	80191488	17214285	2604960	263530	16000	441									7517961216
11	1	176	8019	176000	2311375	20143728	123262293	546957312	1792005930	4374172000	7981924478	10898281728	11148307742	8573377120	4977794250	2187829248	726995973	181293552	33376255	4400000	392931	21296	529							53573031936
12	1	192	9612	233728	3433650	33853248	237463996	1226750976	4764737439	14078204800	31829864088	55193688576	73457993596	75124742784	59187763800	36040204288	17000112591	6210426816	1749479644	376147200	60569586	7070272	564972	27648	625					376583344128
13	1	208	11349	302848	4921150	54156960	428506078	2527562752	11367706779	39524570800	107086069519	226986978816	377018285332	491025029824	501947421300	403532406784	255767554471	128059609392	50663483299	15797267200	3856554702	728110240	104131534	10902528	788125	35152	729			2616492367872
14	1	224	13230	384384	6845475	83184192	733006092	4862567424	24871839993	99610711200	315366381330	793667452032	1592162540787	2548987432320	3258601612200	3329289707520	2722692155547	1785751767072	940886476530	398442844800	135413351073	36773166144	7913473932	1330947072	171136875	16240224	1071630	43904	841	17997636354048

Technical details

Node id: n-4b769f167be1b9f8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d7a387249c78d2a7

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,  16,    9;
  2: 1,  32,  162,   128,     25;
  3: 1,  48,  459,  1408,   1275,     432,       49;
  4: 1,  64,  900,  4864,  11350,   10944,     4900,     1024,        81;
  5: 1,  80, 1485, 11520,  44250,   87264,    86730,    46080,     13365,      2000,       121;
  6: 1,  96, 2214, 22400, 120375,  368064,   648564,   654336,    390015,    140000,     29766,      3456,      169;
  7: 1, 112, 3087, 38528, 266525, 1111824,  2888403,  4697088,   4774707,   3088400,   1289981,    346752,    57967,     5488,     225;
  8: 1, 128, 4104, 60928, 515900, 2733696,  9464056, 21764096,  33328422,  34006400,  23370424,  10934784,  3487484,   746368,  102600,    8192,    289;
  9: 1, 144, 5265, 90624, 908100, 5842368, 25358676, 76068864, 159153822, 232597600, 237748302, 171154944, 87461556, 31808448, 8172900, 1449984, 169065, 11664, 361;

%F Formula

The row polynomials satisfy (1+(12*n-4)*x+(16*n^2+8*n-6)*x^2+(16*n^2-4*n-4)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (1+(12*n+12)*x+(16*n^2+88*n+51)*x^2+(80*n^2+200*n+88)*x^3+(84*n^2+156*n+51)*x^4+(32*n^2+44*n+12)*x^5+(4*n^2+4*n+1)*x^6)*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[((1 + (12*n + 12)*x + (16*n^2 + 88*n + 51)*x^2 + (80*n^2 + 200*n + 88)*x^3 + (84*n^2 + 156*n + 51)*x^4 + (32*n^2 + 44*n + 12)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(1 + (12*n - 4)*x + (16*n^2 + 8*n - 6)*x^2 + (16*n^2 - 4*n - 4)*x^3 + (4*n^2 - 4*n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (12*n + 12)*x + (16*n^2 + 88*n + 51)*x^2 + (80*n^2 + 200*n + 88)*x^3 + (84*n^2 + 156*n + 51)*x^4 + (32*n^2 + 44*n + 12)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(1 + (12*n - 4)*x + (16*n^2 + 8*n - 6)*x^2 + (16*n^2 - 4*n - 4)*x^3 + (4*n^2 - 4*n + 1)*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 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.