A064861 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32 + 92\,x + 96\,x^{2} + 44\,x^{3} + 8\,x^{4} + 24\,n x + 59\,n x^{2} + 46\,n x^{3} + 12\,n x^{4} + 9\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(-32 - 148\,x - 272\,x^{2} - 242\,x^{3} - 102\,x^{4} - 16\,x^{5} - 96\,n - 456\,n x - 803\,n x^{2} - 679\,n x^{3} - 278\,n x^{4} - 44\,n x^{5} - 108\,n^{2} x - 330\,n^{2} x^{2} - 379\,n^{2} x^{3} - 192\,n^{2} x^{4} - 36\,n^{2} x^{5} - 27\,n^{3} x^{2} - 54\,n^{3} x^{3} - 36\,n^{3} x^{4} - 8\,n^{3} x^{5}\right)\,P_{n-1}(x) + \left(64 + 368\,x + 840\,x^{2} + 984\,x^{3} + 628\,x^{4} + 208\,x^{5} + 28\,x^{6} + 96\,n x + 412\,n x^{2} + 682\,n x^{3} + 546\,n x^{4} + 212\,n x^{5} + 32\,n x^{6} + 36\,n^{2} x^{2} + 102\,n^{2} x^{3} + 106\,n^{2} x^{4} + 48\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2 + 6\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	2	6	3															11
1	4	16	15	4														39
2	4	24	39	24	5													96
3	8	56	114	100	40	6												324
4	8	72	198	252	165	54	7											756
5	16	160	504	768	645	306	77	8										2484
6	16	192	744	1440	1605	1080	434	96	9									5616
7	32	416	1776	3872	5010	4086	2128	688	126	10								18144
8	32	480	2400	6240	9850	10098	6895	3120	900	150	11							40176
9	64	1024	5520	15680	27500	32016	25571	14120	5310	1300	187	12						128304
10	64	1152	7056	23040	47100	65016	62923	43344	21195	7200	1617	216	13					279936
11	128	2432	15840	55488	123000	186552	201698	158600	91152	37950	11154	2196	260	14				886464
12	128	2688	19488	76608	192360	334152	419342	389112	269577	139230	52899	14364	2639	294	15			1912896
13	256	5632	43008	179200	480480	899136	1228528	1257472	976905	577990	258951	86436	20839	3430	345	16		6018624
14	256	6144	51456	236544	704480	1475712	2277520	2661504	2391561	1663440	894740	368928	114478	25872	4020	384	17	12877056

Technical details

Node id: n-63edb3976e7cb5f7 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b1698b026521b7b1

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:  2,    6,    3;
  1:  4,   16,   15,     4;
  2:  4,   24,   39,    24,     5;
  3:  8,   56,  114,   100,    40,     6;
  4:  8,   72,  198,   252,   165,    54,     7;
  5: 16,  160,  504,   768,   645,   306,    77,     8;
  6: 16,  192,  744,  1440,  1605,  1080,   434,    96,    9;
  7: 32,  416, 1776,  3872,  5010,  4086,  2128,   688,  126,   10;
  8: 32,  480, 2400,  6240,  9850, 10098,  6895,  3120,  900,  150,  11;
  9: 64, 1024, 5520, 15680, 27500, 32016, 25571, 14120, 5310, 1300, 187, 12;

%F Formula

The row polynomials satisfy (32+(24*n+92)*x+(9*n^2+59*n+96)*x^2+(12*n^2+46*n+44)*x^3+(4*n^2+12*n+8)*x^4)*P(n,x) = -(96*n+32+(108*n^2+456*n+148)*x+(27*n^3+330*n^2+803*n+272)*x^2+(54*n^3+379*n^2+679*n+242)*x^3+(36*n^3+192*n^2+278*n+102)*x^4+(8*n^3+36*n^2+44*n+16)*x^5)*P(n-1,x) + (64+(96*n+368)*x+(36*n^2+412*n+840)*x^2+(102*n^2+682*n+984)*x^3+(106*n^2+546*n+628)*x^4+(48*n^2+212*n+208)*x^5+(8*n^2+32*n+28)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 6*x + 3*x^2.

%t Mathematica

P[0, x_] := 2 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(96*n + 32 + (108*n^2 + 456*n + 148)*x + (27*n^3 + 330*n^2 + 803*n + 272)*x^2 + (54*n^3 + 379*n^2 + 679*n + 242)*x^3 + (36*n^3 + 192*n^2 + 278*n + 102)*x^4 + (8*n^3 + 36*n^2 + 44*n + 16)*x^5)*P[n - 1, x] + (64 + (96*n + 368)*x + (36*n^2 + 412*n + 840)*x^2 + (102*n^2 + 682*n + 984)*x^3 + (106*n^2 + 546*n + 628)*x^4 + (48*n^2 + 212*n + 208)*x^5 + (8*n^2 + 32*n + 28)*x^6)*D[P[n - 1, x], x])/(32 + (24*n + 92)*x + (9*n^2 + 59*n + 96)*x^2 + (12*n^2 + 46*n + 44)*x^3 + (4*n^2 + 12*n + 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 6*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(96*n + 32 + (108*n^2 + 456*n + 148)*x + (27*n^3 + 330*n^2 + 803*n + 272)*x^2 + (54*n^3 + 379*n^2 + 679*n + 242)*x^3 + (36*n^3 + 192*n^2 + 278*n + 102)*x^4 + (8*n^3 + 36*n^2 + 44*n + 16)*x^5)*v[i - 1] + (64 + (96*n + 368)*x + (36*n^2 + 412*n + 840)*x^2 + (102*n^2 + 682*n + 984)*x^3 + (106*n^2 + 546*n + 628)*x^4 + (48*n^2 + 212*n + 208)*x^5 + (8*n^2 + 32*n + 28)*x^6)*deriv(v[i - 1], x))/(32 + (24*n + 92)*x + (9*n^2 + 59*n + 96)*x^2 + (12*n^2 + 46*n + 44)*x^3 + (4*n^2 + 12*n + 8)*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 reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
    
    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 reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
    no recorded property covers this requirement
    
    Parent-row provers of polytool 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.