A257624 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 3\,n\right)\,P_{n}(x) = \left(-50 + 20\,x + 10\,x^{2} - 30\,n - 48\,n x - 9\,n x^{2} - 36\,n^{2} x - 54\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 30\,x^{3} - 18\,n x - 45\,n x^{2} + 63\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(20 + 8\,x + 21\,n + 24\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(6\,x - 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5 + 10\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	10														15
2	25	160	75													260
3	125	1830	2745	500												5200
4	625	18140	60390	36280	3125											118560
5	3125	166370	1045230	1393640	415925	18750										3043040
6	15625	1455960	15739965	39021920	26233275	4367880	109375									86944000
7	78125	12366430	216818415	903079420	1128849275	433636830	43282505	625000								2738736000
8	390625	102993940	2811644400	18424936280	38380875350	27637404420	6560503600	411975760	3515625							94334240000
9	1953125	846607770	34944852060	344172869520	1113098287950	1335717945540	602302521660	93186272160	3809734965	19531250						3528100576000
10	9765625	6897862160	421220810655	6029841711360	28817640893850	53428717821600	40344697251390	12059683422720	1263662431965	34489310800	107421875					142406968704000
11	48828125	55866491030	4964526300885	100704988637700	685869750814650	1863941245102260	2174598119285970	1097391601303440	226586224434825	16548421002950	307265700665	585937500				6170968643840000
12	244140625	450642865740	57542780088810	1621689629765560	15310341601965675	58677961127462280	100031513487154620	78237281503283040	27558614883538215	4054224074413900	210990193658970	2703857194440	3173828125			285763317199360000
13	1220703125	3625162457170	658657224324090	25388984554195640	325143392424038875	1706359110297651090	4080622964404245420	4663569102176280480	2559538665446476635	650286784848077750	69819707524038010	2634628897296360	23563555971605	17089843750		14084049204825600000
14	6103515625	29108721532360	7468176958680945	388817749791159520	6638205745454720125	46612688475036114600	151586144746368173205	242505593313166584960	194896471816759079835	77687814125060191000	14604052640000384275	1166453249373478560	32362100154284095	203761050726520	91552734375	736126305105551360000

Technical details

Node id: n-eab9f49247beab92 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7ef0268628dd91a

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:       5,        10;
  2:      25,       160,          75;
  3:     125,      1830,        2745,          500;
  4:     625,     18140,       60390,        36280,          3125;
  5:    3125,    166370,     1045230,      1393640,        415925,         18750;
  6:   15625,   1455960,    15739965,     39021920,      26233275,       4367880,       109375;
  7:   78125,  12366430,   216818415,    903079420,    1128849275,     433636830,     43282505,      625000;
  8:  390625, 102993940,  2811644400,  18424936280,   38380875350,   27637404420,   6560503600,   411975760,    3515625;
  9: 1953125, 846607770, 34944852060, 344172869520, 1113098287950, 1335717945540, 602302521660, 93186272160, 3809734965, 19531250;

%F Formula

The row polynomials satisfy (3*n+2)*P(n,x) = -(30*n+50+(36*n^2+48*n-20)*x+(27*n^3+54*n^2+9*n-10)*x^2)*P(n-2,x) - ((18*n+30)*x+(27*n^2+45*n)*x^2-(27*n^2+63*n+30)*x^3)*d/dx P(n-2,x) + (21*n+20+(18*n^2+24*n+8)*x)*P(n-1,x) + ((9*n+6)*x-(9*n+6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*P[n - 2, x] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*D[P[n - 2, x], x] + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*P[n - 1, x] + ((9*n + 6)*x - (9*n + 6)*x^2)*D[P[n - 1, x], x])/(3*n + 2)]];
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] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*v[i - 2] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*deriv(v[i - 2], x) + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*v[i - 1] + ((9*n + 6)*x - (9*n + 6)*x^2)*deriv(v[i - 1], x))/(3*n + 2)); 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) = 3*x + 5.?
  - 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) = 3*x + 5.?
    
    ↑ 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.