A253283 → D

nonnegative coefficients

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

Recurrence

\(\left(126 - 253\,n + 167\,n^{2} - 44\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-300 + 455\,n - 239\,n^{2} + 52\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(1104 + 2160\,x - 1684\,n - 3324\,n x + 896\,n^{2} + 1784\,n^{2} x - 200\,n^{3} - 400\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-3}(x) + \left(-1458 - 3888\,x - 3780\,x^{2} + 2268\,n + 6048\,n x + 5952\,n x^{2} - 1242\,n^{2} - 3312\,n^{2} x - 3296\,n^{2} x^{2} + 288\,n^{3} + 768\,n^{3} x + 768\,n^{3} x^{2} - 24\,n^{4} - 64\,n^{4} x - 64\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(792 + 1512\,x - 1292\,n - 2532\,n x + 752\,n^{2} + 1496\,n^{2} x - 184\,n^{3} - 368\,n^{3} x + 16\,n^{4} + 32\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1\)

\(P_{3}(x) = 2 + 6\,x\)

\(P_{4}(x) = 3 + 24\,x + 30\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	2	6													8
4	3	24	30												57
5	4	60	180	140											384
6	5	120	630	1120	630										2505
7	6	210	1680	5040	6300	2772									16008
8	7	336	3780	16800	34650	33264	12012								100849
9	8	504	7560	46200	138600	216216	168168	51480							628736
10	9	720	13860	110880	450450	1009008	1261260	823680	218790						3888657
11	10	990	23760	240240	1261260	3783780	6726720	7001280	3938220	923780					23900040
12	11	1320	38610	480480	3153150	12108096	28588560	42007680	37413090	18475600	3879876				146146473
13	12	1716	60060	900900	7207200	34306272	102918816	199536480	249420600	193993800	85357272	16224936			889928064
14	13	2184	90090	1601600	15315300	88216128	325909584	798145920	1309458150	1422621200	981608628	389398464	67603900		5399971161
15	14	2730	131040	2722720	30630600	209513304	931170240	2793510720	5761615860	8180071900	7852869024	4867480800	1757701400	280816200	32668236552

Technical details

Node id: n-b2ec236586840884 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-883c9e310e304e0b

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: 1;
   3: 2,   6;
   4: 3,  24,    30;
   5: 4,  60,   180,    140;
   6: 5, 120,   630,   1120,    630;
   7: 6, 210,  1680,   5040,   6300,    2772;
   8: 7, 336,  3780,  16800,  34650,   33264,   12012;
   9: 8, 504,  7560,  46200, 138600,  216216,  168168,  51480;
  10: 9, 720, 13860, 110880, 450450, 1009008, 1261260, 823680, 218790;

%F Formula

The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+896*n^2-1684*n+1104+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1242*n^2-2268*n+1458+(64*n^4-768*n^3+3312*n^2-6048*n+3888)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+752*n^2-1292*n+792+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 2 + 6*x, P(4,x) = 3 + 24*x + 30*x^2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 2 + 6*x;
P[4, x_] := 3 + 24*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 2 + 6*x; v[4] = 3 + 24*x + 30*x^2; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 896*n^2 - 1684*n + 1104 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1242*n^2 - 2268*n + 1458 + (64*n^4 - 768*n^3 + 3312*n^2 - 6048*n + 3888)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 752*n^2 - 1292*n + 792 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*v[i - 1])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); 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: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

References

[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.