A274496 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 7\,x + x^{2} - 6\,x^{3} + 10\,n x + 6\,n x^{2} - 3\,n x^{3} + 14\,n^{2} x^{2} + 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(8 - 50\,x - 110\,x^{2} - 116\,x^{3} - 40\,x^{4} - 16\,x^{5} + 84\,n x - 36\,n x^{2} - 138\,n x^{3} - 124\,n x^{4} - 32\,n x^{5} + 188\,n^{2} x^{2} + 126\,n^{2} x^{3} + 18\,n^{2} x^{4} - 4\,n^{2} x^{5} + 144\,n^{3} x^{3} + 112\,n^{3} x^{4} + 28\,n^{3} x^{5} + 42\,n^{4} x^{4} + 20\,n^{4} x^{5} + 4\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-16\,x - 72\,x^{2} - 136\,x^{3} - 152\,x^{4} - 96\,x^{5} - 32\,x^{6} - 68\,n x^{2} - 276\,n x^{3} - 412\,n x^{4} - 312\,n x^{5} - 96\,n x^{6} - 80\,n^{2} x^{3} - 276\,n^{2} x^{4} - 284\,n^{2} x^{5} - 104\,n^{2} x^{6} - 32\,n^{3} x^{4} - 88\,n^{3} x^{5} - 48\,n^{3} x^{6} - 4\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	32							36
3	8	64							72
4	16	256	576						848
5	32	512	1152						1696
6	64	1536	6912	8192					16704
7	128	3072	13824	16384					33408
8	256	8192	55296	131072	102400				297216
9	512	16384	110592	262144	204800				594432
10	1024	40960	368640	1310720	2048000	1179648			4948992
11	2048	81920	737280	2621440	4096000	2359296			9897984
12	4096	196608	2211840	10485760	24576000	28311552	12845056		78630912
13	8192	393216	4423680	20971520	49152000	56623104	25690112		157261824
14	16384	917504	12386304	73400320	229376000	396361728	359661568	134217728	1206337536

Technical details

Node id: n-8ecfb16735d608fb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5367d60e6d4311ec

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:   4,    32;
  3:   8,    64;
  4:  16,   256,    576;
  5:  32,   512,   1152;
  6:  64,  1536,   6912,   8192;
  7: 128,  3072,  13824,  16384;
  8: 256,  8192,  55296, 131072, 102400;
  9: 512, 16384, 110592, 262144, 204800;

%F Formula

The row polynomials satisfy (4+(10*n+7)*x+(14*n^2+6*n+1)*x^2+(6*n^3+11*n^2-3*n-6)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (8+(84*n-50)*x+(188*n^2-36*n-110)*x^2+(144*n^3+126*n^2-138*n-116)*x^3+(42*n^4+112*n^3+18*n^2-124*n-40)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (16*x+(68*n+72)*x^2+(80*n^2+276*n+136)*x^3+(32*n^3+276*n^2+412*n+152)*x^4+(4*n^4+88*n^3+284*n^2+312*n+96)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*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[((8 + (84*n - 50)*x + (188*n^2 - 36*n - 110)*x^2 + (144*n^3 + 126*n^2 - 138*n - 116)*x^3 + (42*n^4 + 112*n^3 + 18*n^2 - 124*n - 40)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (16*x + (68*n + 72)*x^2 + (80*n^2 + 276*n + 136)*x^3 + (32*n^3 + 276*n^2 + 412*n + 152)*x^4 + (4*n^4 + 88*n^3 + 284*n^2 + 312*n + 96)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + 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] = ((8 + (84*n - 50)*x + (188*n^2 - 36*n - 110)*x^2 + (144*n^3 + 126*n^2 - 138*n - 116)*x^3 + (42*n^4 + 112*n^3 + 18*n^2 - 124*n - 40)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*v[i - 1] - (16*x + (68*n + 72)*x^2 + (80*n^2 + 276*n + 136)*x^3 + (32*n^3 + 276*n^2 + 412*n + 152)*x^4 + (4*n^4 + 88*n^3 + 284*n^2 + 312*n + 96)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + 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_A013611 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

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.