A133080 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8 - 16\,x + 32\,x^{2} + 40\,n x - 144\,n x^{2} + 112\,n^{2} x^{2} - 128\,n^{2} x^{3} + 32\,n^{2} x^{4} + 96\,n^{3} x^{3} - 64\,n^{3} x^{4} + 32\,n^{4} x^{4}\right)\,P_{n}(x) = \left(40 - 56\,x - 32\,x^{2} - 448\,x^{3} - 256\,x^{4} + 128\,x^{5} - 24\,n + 372\,n x - 176\,n x^{2} - 288\,n x^{3} - 512\,n x^{4} - 64\,n x^{5} - 204\,n^{2} x + 880\,n^{2} x^{2} + 768\,n^{2} x^{3} + 96\,n^{2} x^{4} - 256\,n^{2} x^{5} - 480\,n^{3} x^{2} + 288\,n^{3} x^{3} + 800\,n^{3} x^{4} + 128\,n^{3} x^{5} - 384\,n^{4} x^{3} - 288\,n^{4} x^{4} + 192\,n^{4} x^{5} - 96\,n^{5} x^{4} - 128\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(3 - 8\,x - 44\,x^{2} - 96\,x^{3} - 144\,x^{4} + 64\,x^{6} + 36\,n x + 4\,n x^{2} - 240\,n x^{3} - 304\,n x^{4} - 192\,n x^{5} + 132\,n^{2} x^{2} + 320\,n^{2} x^{3} + 32\,n^{2} x^{4} - 224\,n^{2} x^{5} - 128\,n^{2} x^{6} + 144\,n^{3} x^{3} + 448\,n^{3} x^{4} + 320\,n^{3} x^{5} + 48\,n^{4} x^{4} + 160\,n^{4} x^{5} + 128\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	1	16	36													53
3	2	40	144	128												314
4	1	32	216	512	400											1161
5	2	72	576	1792	2400	1152										5994
6	1	48	540	2560	6000	6912	3136									19197
7	2	104	1296	7040	20000	31104	25088	8192								92826
8	1	64	1008	7168	28000	64512	87808	65536	20736							274833
9	2	136	2304	17920	78400	209664	351232	360448	207360	51200						1278666
10	1	80	1620	15360	84000	290304	658560	983040	933120	512000	123904					3601989
11	2	168	3600	36480	216000	822528	2107392	3686400	4354560	3328000	1486848	294912				16336890
12	1	96	2376	28160	198000	912384	2897664	6488064	10264320	11264000	8177664	3538944	692224			44463897
13	2	200	5184	64768	484000	2395008	8279040	20545536	36951552	47872000	43614208	26542080	9691136	1605632		198050346
14	1	112	3276	46592	400400	2306304	9417408	28114944	62270208	102502400	124027904	107347968	62992384	22478848	3686400	525595149

Technical details

Node id: n-45e5afb1a3ebeb9e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-a3f2ba5016995d8c

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 2,   8;
  2: 1,  16,   36;
  3: 2,  40,  144,   128;
  4: 1,  32,  216,   512,   400;
  5: 2,  72,  576,  1792,  2400,   1152;
  6: 1,  48,  540,  2560,  6000,   6912,   3136;
  7: 2, 104, 1296,  7040, 20000,  31104,  25088,   8192;
  8: 1,  64, 1008,  7168, 28000,  64512,  87808,  65536,  20736;
  9: 2, 136, 2304, 17920, 78400, 209664, 351232, 360448, 207360, 51200;

%F Formula

The row polynomials satisfy (8+(40*n-16)*x+(112*n^2-144*n+32)*x^2+(96*n^3-128*n^2)*x^3+(32*n^4-64*n^3+32*n^2)*x^4)*P(n,x) = -(24*n-40+(204*n^2-372*n+56)*x+(480*n^3-880*n^2+176*n+32)*x^2+(384*n^4-288*n^3-768*n^2+288*n+448)*x^3+(96*n^5+288*n^4-800*n^3-96*n^2+512*n+256)*x^4+(128*n^5-192*n^4-128*n^3+256*n^2+64*n-128)*x^5)*P(n-1,x) + (3+(36*n-8)*x+(132*n^2+4*n-44)*x^2+(144*n^3+320*n^2-240*n-96)*x^3+(48*n^4+448*n^3+32*n^2-304*n-144)*x^4+(160*n^4+320*n^3-224*n^2-192*n)*x^5+(128*n^4-128*n^2+64)*x^6)*d/dx 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[(-(24*n - 40 + (204*n^2 - 372*n + 56)*x + (480*n^3 - 880*n^2 + 176*n + 32)*x^2 + (384*n^4 - 288*n^3 - 768*n^2 + 288*n + 448)*x^3 + (96*n^5 + 288*n^4 - 800*n^3 - 96*n^2 + 512*n + 256)*x^4 + (128*n^5 - 192*n^4 - 128*n^3 + 256*n^2 + 64*n - 128)*x^5)*P[n - 1, x] + (3 + (36*n - 8)*x + (132*n^2 + 4*n - 44)*x^2 + (144*n^3 + 320*n^2 - 240*n - 96)*x^3 + (48*n^4 + 448*n^3 + 32*n^2 - 304*n - 144)*x^4 + (160*n^4 + 320*n^3 - 224*n^2 - 192*n)*x^5 + (128*n^4 - 128*n^2 + 64)*x^6)*D[P[n - 1, x], x])/(8 + (40*n - 16)*x + (112*n^2 - 144*n + 32)*x^2 + (96*n^3 - 128*n^2)*x^3 + (32*n^4 - 64*n^3 + 32*n^2)*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] = (-(24*n - 40 + (204*n^2 - 372*n + 56)*x + (480*n^3 - 880*n^2 + 176*n + 32)*x^2 + (384*n^4 - 288*n^3 - 768*n^2 + 288*n + 448)*x^3 + (96*n^5 + 288*n^4 - 800*n^3 - 96*n^2 + 512*n + 256)*x^4 + (128*n^5 - 192*n^4 - 128*n^3 + 256*n^2 + 64*n - 128)*x^5)*v[i - 1] + (3 + (36*n - 8)*x + (132*n^2 + 4*n - 44)*x^2 + (144*n^3 + 320*n^2 - 240*n - 96)*x^3 + (48*n^4 + 448*n^3 + 32*n^2 - 304*n - 144)*x^4 + (160*n^4 + 320*n^3 - 224*n^2 - 192*n)*x^5 + (128*n^4 - 128*n^2 + 64)*x^6)*deriv(v[i - 1], x))/(8 + (40*n - 16)*x + (112*n^2 - 144*n + 32)*x^2 + (96*n^3 - 128*n^2)*x^3 + (32*n^4 - 64*n^3 + 32*n^2)*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 basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    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

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.