type_b_eulerian → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-1 - 60\,x - 15\,x^{2} + 40\,n x + 16\,n x^{2} + 4\,n^{2} + 48\,n^{2} x + 152\,n^{2} x^{2} - 32\,n^{3} x - 320\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(16\,x - 60\,x^{2} - 20\,x^{3} + 8\,n x^{2} + 8\,n x^{3} - 16\,n^{2} x + 48\,n^{2} x^{2} + 224\,n^{2} x^{3} + 32\,n^{3} x^{2} - 352\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(12\,x^{2} - 8\,x^{3} - 4\,x^{4} - 16\,n^{2} x^{2} - 32\,n^{2} x^{3} + 48\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(2 - 4\,x - 4\,n + 24\,n x - 48\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2} - 8\,n x + 8\,n x^{2} + 16\,n^{2} x + 16\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	24	9													34
3	1	92	207	16												316
4	1	304	2070	1216	25											3616
5	1	948	15138	26912	5925	36										48960
6	1	2888	94887	376768	263575	25992	49									764160
7	1	8716	545913	4155568	6493075	2183652	106771	64								13493760
8	1	26208	2984508	39764608	116875350	89470368	16248988	419328	81							265789440
9	1	78692	15807060	347327552	1735332350	2498878584	1063690628	112405760	1593513	100						5775114240
10	1	236152	82045269	2852814464	22673082050	54975328848	44439240818	11411257856	738407421	5903800	121					137178316800
11	1	708540	420321843	22449291600	270920577450	1029168713304	1400812970886	693556678272	113649538725	4670242700	21433335	144				3535670476800
12	1	2125712	2135087730	171342579008	3034594505175	17172740319264	36421137243036	30529316123136	9832086196767	1070891118800	28705068370	76525632	169			98263026892800
13	1	6377236	10785444246	1279107994976	32398083360875	263187016998588	824084899011108	1076355786463488	592170788246823	129592333443500	9673254212006	172567107936	269438221	196		2928924898099200
14	1	19131816	54285940755	9394760796224	333553481333025	3781468747738008	16802958830472843	32290673593904640	27776319699353067	10504079854827800	1614398849651841	84552847166016	1019369331955	937458984	225	93198475277107200

Technical details

Node id: n-cbf7604ef9181b71 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5d3f47477444bdf9

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: 1,     4;
  2: 1,    24,        9;
  3: 1,    92,      207,        16;
  4: 1,   304,     2070,      1216,         25;
  5: 1,   948,    15138,     26912,       5925,         36;
  6: 1,  2888,    94887,    376768,     263575,      25992,         49;
  7: 1,  8716,   545913,   4155568,    6493075,    2183652,     106771,        64;
  8: 1, 26208,  2984508,  39764608,  116875350,   89470368,   16248988,    419328,      81;
  9: 1, 78692, 15807060, 347327552, 1735332350, 2498878584, 1063690628, 112405760, 1593513, 100;

%F Formula

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2-1-(32*n^3-48*n^2-40*n+60)*x+(144*n^4-320*n^3+152*n^2+16*n-15)*x^2)*P(n-2,x) - ((16*n^2-16)*x-(32*n^3+48*n^2+8*n-60)*x^2-(64*n^4-352*n^3+224*n^2+8*n-20)*x^3)*d/dx P(n-2,x) - ((16*n^2-12)*x^2-(64*n^3-32*n^2-8)*x^3+(64*n^3-48*n^2+4)*x^4)*d^2/dx^2 P(n-2,x) - (4*n-2-(32*n^3-48*n^2+24*n-4)*x)*P(n-1,x) + ((16*n^2-8*n-4)*x-(32*n^3-16*n^2-8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 1 - (32*n^3 - 48*n^2 - 40*n + 60)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*P[n - 2, x] - ((16*n^2 - 16)*x - (32*n^3 + 48*n^2 + 8*n - 60)*x^2 - (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 12)*x^2 - (64*n^3 - 32*n^2 - 8)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*P[n - 1, x] + ((16*n^2 - 8*n - 4)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 1 - (32*n^3 - 48*n^2 - 40*n + 60)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*v[i - 2] - ((16*n^2 - 16)*x - (32*n^3 + 48*n^2 + 8*n - 60)*x^2 - (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 12)*x^2 - (64*n^3 - 32*n^2 - 8)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*v[i - 1] + ((16*n^2 - 8*n - 4)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 type_b_eulerian?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from type_b_eulerian?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.