A225473 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-21 - 48\,x - 32\,x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 28\,x^{2} - 16\,x^{3}\right)\,P_{n-2}'(x) + \left(10 + 16\,x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	4														4
3	40	64													104
4	316	1344	1152												2812
5	2320	19840	41472	24576											88208
6	16564	255360	979200	1351680	614400										3217204
7	116920	3073984	19284480	46571520	47923200	17694720									134664824
8	821356	35684544	344598912	1296261120	2266521600	1857945600	578027520								6379860652
9	5758240	405671680	5811305472	31980404736	84536524800	115440353280	78611742720	21139292160							337931053088
10	40333924	4554520320	94471672320	731602206720	2743850188800	5551541452800	6208015564800	3614818959360	856141332480						19805036231524
11	282416200	50745066304	1499056450560	15915837603840	81398609510400	228914504663040	373798836633600	353554661376000	179789679820800	38050725888000					1272972939428744
12	1977149596	562714388544	23399257951872	334380785418240	2270063958835200	8524971579801600	19065637754634240	26044770602188800	21330761298739200	9626833649664000	1841655132979200				89063038711750492
13	13840755760	6221492667520	361117728271872	6852419092586496	60570990688665600	295552074669096960	869517175333847040	1608824580388945920	1886712900039475200	1361987682435072000	552496539893760000	96435759690547200			6739317475293691568
14	96887416084	68657871434880	5528752792093440	137899807628943360	1564443263153971200	9724350547899187200	36609733541206425600	88108345582965227520	138739198607291842560	142419377013129216000	91887541204864204800	33848951651382067200	5432547795900825600		548477986522972855444
15	678218289880	756786784441024	84167133567229440	2738043071181250560	39429690243264921600	307613430772112424960	1452563857675439308800	4422662350625046528000	8979216741794473574400	12290219816333082624000	11211812284064071680000	6537862328220647424000	2205614405135735193600	327624420922019020800	47777442293455643911064

Technical details

Node id: n-c5e19c36ca165482 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0b6e449e7f11e863

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:        4;
   3:       40,         64;
   4:      316,       1344,        1152;
   5:     2320,      19840,       41472,        24576;
   6:    16564,     255360,      979200,      1351680,        614400;
   7:   116920,    3073984,    19284480,     46571520,      47923200,      17694720;
   8:   821356,   35684544,   344598912,   1296261120,    2266521600,    1857945600,     578027520;
   9:  5758240,  405671680,  5811305472,  31980404736,   84536524800,  115440353280,   78611742720,   21139292160;
  10: 40333924, 4554520320, 94471672320, 731602206720, 2743850188800, 5551541452800, 6208015564800, 3614818959360, 856141332480;

%F Formula

The row polynomials satisfy P(n,x) = -(21+48*x+32*x^2)*P(n-2,x) - (12*x+28*x^2+16*x^3)*d/dx P(n-2,x) + (10+16*x)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(21 + 48*x + 32*x^2)*P[n - 2, x] - (12*x + 28*x^2 + 16*x^3)*D[P[n - 2, x], x] + (10 + 16*x)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = -(21 + 48*x + 32*x^2)*v[i - 2] - (12*x + 28*x^2 + 16*x^3)*deriv(v[i - 2], x) + (10 + 16*x)*v[i - 1] + (4*x + 4*x^2)*deriv(v[i - 1], 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, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.