A257613 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n + n^{2}\right)\,P_{n}(x) = \left(-16 - 24\,n - 80\,n x + 64\,n x^{2} - 8\,n^{2} - 96\,n^{2} x + 164\,n^{2} x^{2} - 32\,n^{3} x + 136\,n^{3} x^{2} + 36\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-44\,x + 120\,x^{2} - 92\,x^{3} - 48\,n x + 212\,n x^{2} - 208\,n x^{3} - 16\,n^{2} x + 144\,n^{2} x^{2} - 124\,n^{2} x^{3} + 32\,n^{3} x^{2} + 8\,n^{3} x^{3} + 16\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-12\,x^{2} + 40\,x^{3} - 28\,x^{4} - 12\,n x^{2} + 84\,n x^{3} - 72\,n x^{4} - 4\,n^{2} x^{2} + 64\,n^{2} x^{3} - 60\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(8 + 8\,x + 14\,n + 24\,n x + 6\,n^{2} + 24\,n^{2} x + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 16\,x^{2} + 10\,n x - 40\,n x^{2} + 4\,n^{2} x - 32\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 16\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	16														20
2	16	192	144													352
3	64	1664	3744	1024												6496
4	256	12544	59904	50176	6400											129280
5	1024	87552	761472	1353728	547200	36864										2787840
6	4096	582656	8455680	27074560	23488000	5243904	200704									65049600
7	16384	3758080	85999104	451133440	704896000	343996416	46036480	1048576								1636884480
8	65536	23728128	823283712	6651756544	16917504000	14966452224	4482322432	379650048	5308416							44250071040
9	262144	147611648	7547258880	89935413248	348517222400	501864800256	275427203072	53669396480	2989135872	26214400						1280124518400
10	1048576	908738560	67020595200	1140866416640	6430592000000	14052214407168	12603960320000	4563465666560	603185356800	22718464000	126877696					39485059891200
11	4194304	5553094656	581147320320	13791618662400	109253971968000	344891841380352	469436117434368	279690168238080	69820069478400	6457192448000	167981113344	603979776				1294096269312000
12	16777216	33754775552	4949045673984	160645504040960	1742035746816000	7660348732735488	15021955757899776	13618397747085312	5644195819683840	1004034400256000	66537169616896	1215171919872	2835349504			44924351702630400
13	67108864	204407701504	41567019761664	1817614322499584	26426618163200000	157415511766597632	428029836078415872	559059377735073792	354184901474844672	105706472652800000	13745708313903104	665072316186624	8636225388544	13153337344		1647101533696819200
14	268435456	1234499272704	345413843288064	20097463249534976	385279955052134400	3041012428142542848	11133677419789811712	20126137598462656512	18404650428632137728	8447256744840396800	1864754982452330496	180877169245814784	6486104390631424	60490464362496	60397977600	63610637493731328000

Technical details

Node id: n-c9e46b637e60b92e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5cf267279c5da1ec

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:      4,        16;
  2:     16,       192,        144;
  3:     64,      1664,       3744,        1024;
  4:    256,     12544,      59904,       50176,         6400;
  5:   1024,     87552,     761472,     1353728,       547200,        36864;
  6:   4096,    582656,    8455680,    27074560,     23488000,      5243904,       200704;
  7:  16384,   3758080,   85999104,   451133440,    704896000,    343996416,     46036480,     1048576;
  8:  65536,  23728128,  823283712,  6651756544,  16917504000,  14966452224,   4482322432,   379650048,    5308416;
  9: 262144, 147611648, 7547258880, 89935413248, 348517222400, 501864800256, 275427203072, 53669396480, 2989135872, 26214400;

%F Formula

The row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(8*n^2+24*n+16+(32*n^3+96*n^2+80*n)*x-(36*n^4+136*n^3+164*n^2+64*n)*x^2)*P(n-2,x) - ((16*n^2+48*n+44)*x-(32*n^3+144*n^2+212*n+120)*x^2-(16*n^4+8*n^3-124*n^2-208*n-92)*x^3)*d/dx P(n-2,x) - ((4*n^2+12*n+12)*x^2-(16*n^3+64*n^2+84*n+40)*x^3+(16*n^3+60*n^2+72*n+28)*x^4)*d^2/dx^2 P(n-2,x) + (6*n^2+14*n+8+(8*n^3+24*n^2+24*n+8)*x)*P(n-1,x) + ((4*n^2+10*n+8)*x-(8*n^3+32*n^2+40*n+16)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 + 24*n + 16 + (32*n^3 + 96*n^2 + 80*n)*x - (36*n^4 + 136*n^3 + 164*n^2 + 64*n)*x^2)*P[n - 2, x] - ((16*n^2 + 48*n + 44)*x - (32*n^3 + 144*n^2 + 212*n + 120)*x^2 - (16*n^4 + 8*n^3 - 124*n^2 - 208*n - 92)*x^3)*D[P[n - 2, x], x] - ((4*n^2 + 12*n + 12)*x^2 - (16*n^3 + 64*n^2 + 84*n + 40)*x^3 + (16*n^3 + 60*n^2 + 72*n + 28)*x^4)*D[P[n - 2, x], {x, 2}] + (6*n^2 + 14*n + 8 + (8*n^3 + 24*n^2 + 24*n + 8)*x)*P[n - 1, x] + ((4*n^2 + 10*n + 8)*x - (8*n^3 + 32*n^2 + 40*n + 16)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*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] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 + 24*n + 16 + (32*n^3 + 96*n^2 + 80*n)*x - (36*n^4 + 136*n^3 + 164*n^2 + 64*n)*x^2)*v[i - 2] - ((16*n^2 + 48*n + 44)*x - (32*n^3 + 144*n^2 + 212*n + 120)*x^2 - (16*n^4 + 8*n^3 - 124*n^2 - 208*n - 92)*x^3)*deriv(v[i - 2], x) - ((4*n^2 + 12*n + 12)*x^2 - (16*n^3 + 64*n^2 + 84*n + 40)*x^3 + (16*n^3 + 60*n^2 + 72*n + 28)*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n^2 + 14*n + 8 + (8*n^3 + 24*n^2 + 24*n + 8)*x)*v[i - 1] + ((4*n^2 + 10*n + 8)*x - (8*n^3 + 32*n^2 + 40*n + 16)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.?
    
    ↑ 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.