A304330 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-24\,x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 36\,x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{3} - 8\,x^{4}\right)\,P_{n-2}''(x) + \left(1 + 24\,x\right)\,P_{n-1}(x) + \left(3\,x + 22\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	1	24													25
3	1	120	1080												1201
4	1	504	15120	80640											96265
5	1	2040	158760	2419200	9072000										11652001
6	1	8184	1520640	50561280	498960000	1437004800									1988054905
7	1	32760	14054040	922521600	18162144000	130767436800	305124019200								454990208401
8	1	131064	127960560	15809713920	557837280000	7584511334400	42717362688000	83691159552000							134566808659945
9	1	524280	1157542920	262509811200	15724524816000	361403833190400	3703595345049600	17072996548608000	28810681675776000						49964665595318401
10	1	2097144	10441478880	4286586850560	422645474160000	15501302725708800	258214252488192000	2108515073753088000	8211044277596160000	12164510040883200000					22758211889950935385
11	1	8388600	94067681400	69365020032000	11048377874688000	624993941232460800	15943941650680243200	205769445402673152000	1390950900624789504000	4683336365740032000000	6182004002776842240000				12478640767973220390001
12	1	33554424	846986619600	1116864040723200	284013011620800000	24249844939719168000	913960187738357760000	17542439929957662720000	183503154530478182400000	1055624016837803212800000	3128094025405082173440000	3722690410399436636160000			8108392542266158626145225
13	1	134217720	7624389525480	17933066319168000	7225972495101360000	917982078870624768000	49933099608046566912000	1373402378439811399680000	20902740462856658165760000	183041511374426664960000000	914967502430986535731200000	2419748766759633813504000000	2621394497322936631296000000		6161479279054880218192591201
14	1	536870904	68625545526720	287498348857923840	182666782338440400000	34191948882566547072000	2641640075541144437760000	101593688112640702218240000	2165915746269294385766400000	27129752666204366610432000000	203732763874633001956147200000	899420616604555888479436800000	2146922093307485101031424000000	2134218412281997023510528000000	5413693824184236089700318193465

Technical details

Node id: n-9239b1466464de00 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d46d0fd93c8b8ae2

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 -1)
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: 0;
  1: 1;
  2: 1,     24;
  3: 1,    120,       1080;
  4: 1,    504,      15120,        80640;
  5: 1,   2040,     158760,      2419200,        9072000;
  6: 1,   8184,    1520640,     50561280,      498960000,      1437004800;
  7: 1,  32760,   14054040,    922521600,    18162144000,    130767436800,     305124019200;
  8: 1, 131064,  127960560,  15809713920,   557837280000,   7584511334400,   42717362688000,    83691159552000;
  9: 1, 524280, 1157542920, 262509811200, 15724524816000, 361403833190400, 3703595345049600, 17072996548608000, 28810681675776000;

%F Formula

The row polynomials satisfy P(n,x) = -24*x^2*P(n-2,x) - (4*x^2+36*x^3)*d/dx P(n-2,x) - (2*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (1+24*x)*P(n-1,x) + (3*x+22*x^2)*d/dx P(n-1,x) + (x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-24*x^2*P[n - 2, x] - (4*x^2 + 36*x^3)*D[P[n - 2, x], x] - (2*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 24*x)*P[n - 1, x] + (3*x + 22*x^2)*D[P[n - 1, x], x] + (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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -24*x^2*v[i - 2] - (4*x^2 + 36*x^3)*deriv(v[i - 2], x) - (2*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 24*x)*v[i - 1] + (3*x + 22*x^2)*deriv(v[i - 1], x) + (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 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.?
  - 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.