A257606 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 6\,n + n^{2}\right)\,P_{n}(x) = \left(-144 - 408\,x + 612\,x^{2} - 84\,n - 428\,n x + 876\,n x^{2} - 12\,n^{2} - 144\,n^{2} x + 461\,n^{2} x^{2} - 16\,n^{3} x + 106\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-111\,x + 642\,x^{2} - 171\,x^{3} - 56\,n x + 557\,n x^{2} - 24\,n x^{3} - 8\,n^{2} x + 164\,n^{2} x^{2} + 77\,n^{2} x^{3} + 16\,n^{3} x^{2} + 34\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-15\,x^{2} + 150\,x^{3} - 135\,x^{4} - 7\,n x^{2} + 133\,n x^{3} - 126\,n x^{4} - n^{2} x^{2} + 40\,n^{2} x^{3} - 39\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(72 + 108\,x + 45\,n + 108\,n x + 7\,n^{2} + 36\,n^{2} x + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(24\,x - 144\,x^{2} + 13\,n x - 132\,n x^{2} + 2\,n^{2} x - 40\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	160	144													320
3	64	1184	2664	1024												4936
4	256	7712	31968	30848	6400											77184
5	1024	46752	313272	556928	292200	36864										1247040
6	4096	270624	2721168	7796992	7558800	2435616	200704									20988000
7	16384	1516960	21807144	93280000	145750000	87228576	18582760	1048576								369230400
8	65536	8305696	164974464	1001874304	2332000000	2254217184	898194304	132891136	5308416							6797831040
9	262144	44674208	1195412760	9945999488	32744857400	47152594656	30459623432	8500712960	904652712	26214400						130975004160
10	1048576	237002528	8378680176	92998957056	417365101200	848746703808	818035598352	371995828224	75408121584	5925063200	126877696					2639218982400
11	4194304	1243732896	57204405000	829737876480	4937340384000	13648777791552	18577503105168	12639591383040	4200547999680	635604500000	37622920104	603979776				55565782272000
12	16777216	6470322720	382404016224	7130222495360	55056308256000	201046471806528	371550062103360	357415949878272	178382438749440	44563890596000	5141209551456	232931617920	2835349504			1220911211520000
13	67108864	33425355424	2512797489144	59429168537984	585283110485000	2760791252922432	6725613073914672	8784474218990592	6211780319075472	2341132441940000	449433087068504	40204759826304	1412221266664	13153337344		27962113097318400
14	268435456	171690179872	16280097730128	483012112809728	5982277945648400	35803621104581088	112349054536879776	193258432817793024	185719865663005344	99454503068280800	28954225256938256	4347109015287552	305704057376848	8412818813728	60397977600	666682730851737600

Technical details

Node id: n-26fd66898404a56a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-de45cc49490c28bd

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,      160,        144;
  3:     64,     1184,       2664,       1024;
  4:    256,     7712,      31968,      30848,        6400;
  5:   1024,    46752,     313272,     556928,      292200,       36864;
  6:   4096,   270624,    2721168,    7796992,     7558800,     2435616,      200704;
  7:  16384,  1516960,   21807144,   93280000,   145750000,    87228576,    18582760,    1048576;
  8:  65536,  8305696,  164974464, 1001874304,  2332000000,  2254217184,   898194304,  132891136,   5308416;
  9: 262144, 44674208, 1195412760, 9945999488, 32744857400, 47152594656, 30459623432, 8500712960, 904652712, 26214400;

%F Formula

The row polynomials satisfy (n^2+6*n+9)*P(n,x) = -(12*n^2+84*n+144+(16*n^3+144*n^2+428*n+408)*x-(9*n^4+106*n^3+461*n^2+876*n+612)*x^2)*P(n-2,x) - ((8*n^2+56*n+111)*x-(16*n^3+164*n^2+557*n+642)*x^2-(4*n^4+34*n^3+77*n^2-24*n-171)*x^3)*d/dx P(n-2,x) - ((n^2+7*n+15)*x^2-(4*n^3+40*n^2+133*n+150)*x^3+(4*n^3+39*n^2+126*n+135)*x^4)*d^2/dx^2 P(n-2,x) + (7*n^2+45*n+72+(4*n^3+36*n^2+108*n+108)*x)*P(n-1,x) + ((2*n^2+13*n+24)*x-(4*n^3+40*n^2+132*n+144)*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[(-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*P[n - 2, x] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*D[P[n - 2, x], x] - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*P[n - 1, x] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n + 9)]];
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] = (-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*v[i - 2] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*deriv(v[i - 2], x) - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*v[i - 1] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n + 9)); 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) = 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) = x + 4.?
    
    ↑ Liu–Wang theorem (above)

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.