A151775 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 + 4\,x + 22\,x^{2} + 16\,x^{3} - 8\,x^{4}\right)\,P_{n-2}(x) + \left(6 + 40\,x + 138\,x^{2} + 156\,x^{3} + 12\,x^{4} - 40\,x^{5}\right)\,P_{n-2}'(x) + \left(4\,x + 32\,x^{2} + 60\,x^{3} + 24\,x^{4} - 24\,x^{5} - 16\,x^{6}\right)\,P_{n-2}''(x) + \left(8 - 14\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(6 + 28\,x + 22\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	33	76	24													133
3	723	3302	3480	720												8225
4	25953	191896	386064	262080	40320											906313
5	1376643	14824202	46756560	59300640	29635200	3628800										155522045
6	101031873	1483727716	6631273704	12938016960	11920487040	4710182400	479001600									38263721293
7	9795436563	187528805102	1116344796840	3035876603760	4251296649600	3069402336000	1002550348800	87178291200								12759973267865
8	1212135593793	29265860447536	222494217046944	794354143040640	1533871279100160	1669219346995200	989822318284800	275483400192000	20922789888000							5536645490589073
9	186388033956483	5532759960522002	52101046426636320	234537391905814080	588855455629459200	876964309110489600	779061750321254400	391736348933529600	94968543301632000	6402373705728000						3030346367329021685
10	34859622790687713	1246737856831391356	14206682438609129784	78315489565451349120	245777182630877479680	472657067359647206400	571408739489197209600	428446433418651648000	186948245144973312000	40142883134914560000	2432902008176640000					2041617222670120613653
11	7791941518975112403	330263690217320550902	4470711925426249107000	29518256162845953795600	112552199308318160160000	268859193467501112537600	417554944224982608844800	424223063702947360051200	275014952407658631168000	105965168611233964032000	20419346554626539520000	1124000727777607680000				1660039892725054482559505
12	2051293521728340489633	101634133744314628799176	1610050623350109685186224	12511999302715611365198400	56732509241052227309865600	163706773191725959582464000	314305097727782094506496000	408577206483228320778240000	357662420760895626055680000	204141681227021355417600000	70412276257809714708480000	12305559967709248880640000	620448401733239439360000			1602689708612289550698499033
13	628173356956461494680323	35961506444610162083467802	658432459422184085374160880	5937800623334429963183572320	31481060263614843558204528000	107452628136227607272047008000	248112204803697668681415168000	397230117758105347507663872000	443491621714341539262443520000	340851374547435595806842880000	173729079439019276614041600000	54232619235000292845158400000	8670766414222021165056000000	403291461126605635584000000		1812287586535348979020592857325
14	221398076445213367209575553	14499233101039276290819310996	303611579932660410232283520264	3140279058135983550862297884480	19204078196311337833690422337920	76268201827210671241450783296000	207455746529613581452641412224000	398061047914050659588995184640000	545700714089660104712917340160000	533498591042263472257478246400000	365302770798113407074816000000000	168167316665372960495547187200000	47941151453986906948357324800000	7063246650171371101618176000000	304888344611713860501504000000	2372426364780612315018766191349213

Technical details

Node id: n-fe550175f722890a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ad67aec3cd3f88df

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:               3,                2;
  2:              33,               76,                24;
  3:             723,             3302,              3480,                720;
  4:           25953,           191896,            386064,             262080,              40320;
  5:         1376643,         14824202,          46756560,           59300640,           29635200,            3628800;
  6:       101031873,       1483727716,        6631273704,        12938016960,        11920487040,         4710182400,          479001600;
  7:      9795436563,     187528805102,     1116344796840,      3035876603760,      4251296649600,      3069402336000,      1002550348800,        87178291200;
  8:   1212135593793,   29265860447536,   222494217046944,    794354143040640,   1533871279100160,   1669219346995200,    989822318284800,    275483400192000,    20922789888000;
  9: 186388033956483, 5532759960522002, 52101046426636320, 234537391905814080, 588855455629459200, 876964309110489600, 779061750321254400, 391736348933529600, 94968543301632000, 6402373705728000;

%F Formula

The row polynomials satisfy P(n,x) = -(3-4*x-22*x^2-16*x^3+8*x^4)*P(n-2,x) + (6+40*x+138*x^2+156*x^3+12*x^4-40*x^5)*d/dx P(n-2,x) + (4*x+32*x^2+60*x^3+24*x^4-24*x^5-16*x^6)*d^2/dx^2 P(n-2,x) + (8-14*x^2+4*x^3)*P(n-1,x) + (6+28*x+22*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3 - 4*x - 22*x^2 - 16*x^3 + 8*x^4)*P[n - 2, x] + (6 + 40*x + 138*x^2 + 156*x^3 + 12*x^4 - 40*x^5)*D[P[n - 2, x], x] + (4*x + 32*x^2 + 60*x^3 + 24*x^4 - 24*x^5 - 16*x^6)*D[P[n - 2, x], {x, 2}] + (8 - 14*x^2 + 4*x^3)*P[n - 1, x] + (6 + 28*x + 22*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - 4*x - 22*x^2 - 16*x^3 + 8*x^4)*v[i - 2] + (6 + 40*x + 138*x^2 + 156*x^3 + 12*x^4 - 40*x^5)*deriv(v[i - 2], x) + (4*x + 32*x^2 + 60*x^3 + 24*x^4 - 24*x^5 - 16*x^6)*deriv(deriv(v[i - 2], x), x) + (8 - 14*x^2 + 4*x^3)*v[i - 1] + (6 + 28*x + 22*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*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 - 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - 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.