A156289 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 10\,x - 15\,x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 14\,x^{2} - 6\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 3\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 10\,x\right)\,P_{n-1}(x) + \left(5\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	3														3
3	15	30													45
4	63	420	315												798
5	255	4410	9450	3780											17895
6	1023	42240	197505	207900	51975										500643
7	4095	390390	3603600	7567560	4729725	810810									17106180
8	16383	3554460	61756695	232432200	274324050	113513400	14189175								699786363
9	65535	32153970	1025428950	6551885340	13071608550	9841611780	2894591700	275675400							33693021225
10	262143	290041080	16744479885	176102280900	560666331225	686156070600	357482074950	78567489000	5892561675						1881901591458
11	1048575	2612991150	270957109500	4603490781120	22605394286475	42368042226510	34886584032300	13309332636600	2268636244875	137493105750					120452544462855
12	4194303	23527406100	4362750159075	118338754842000	877092192553500	2428678219968000	2974182117031125	1755852433335000	511350609594825	69571511509500	3478575575475				8742930696168903
13	16777215	211788597930	70051040309250	3010821872958900	33202476811003500	132687870982266600	232849524339432000	200008156802454000	88666406722018125	20349667116528750	2261074124058750	94870242967500			713201131859372520
14	67108863	1906265153520	1123040425226265	76111159307683500	1236687965949310875	7019664316382718000	17224407299913313500	20724594316736310000	13141815023854266750	4531192544613735000	840441251912637375	77698728990382500	2774954606799375		64876512509024645523
15	268435455	17157057470310	17988662587732200	1916255467794802680	45567295214655839625	363255883771562431650	1225206192731149629000	2013337767196814058000	1757169705905665301250	854916648454861492500	235662711654147277500	36052210251537480000	2816578925901365625	86663966950811250	6536005919360954127045

Technical details

Node id: n-9212bcd5fb1e697f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-48c328a3a13ffc6f

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      0;
   2:      3;
   3:     15,        30;
   4:     63,       420,         315;
   5:    255,      4410,        9450,         3780;
   6:   1023,     42240,      197505,       207900,        51975;
   7:   4095,    390390,     3603600,      7567560,      4729725,       810810;
   8:  16383,   3554460,    61756695,    232432200,    274324050,    113513400,     14189175;
   9:  65535,  32153970,  1025428950,   6551885340,  13071608550,   9841611780,   2894591700,   275675400;
  10: 262143, 290041080, 16744479885, 176102280900, 560666331225, 686156070600, 357482074950, 78567489000, 5892561675;

%F Formula

The row polynomials satisfy P(n,x) = -(4+10*x+15*x^2)*P(n-2,x) - (5*x+14*x^2+6*x^3)*d/dx P(n-2,x) - (x^2+3*x^3)*d^2/dx^2 P(n-2,x) + (5+10*x)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(4 + 10*x + 15*x^2)*P[n - 2, x] - (5*x + 14*x^2 + 6*x^3)*D[P[n - 2, x], x] - (x^2 + 3*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 10*x)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = -(4 + 10*x + 15*x^2)*v[i - 2] - (5*x + 14*x^2 + 6*x^3)*deriv(v[i - 2], x) - (x^2 + 3*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 10*x)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.