A357613 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x - 2\,n x^{2} - 2\,n^{2} - 18\,n^{2} x + 4\,n^{2} x^{2} - 8\,n^{3} + 12\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(-18 - 30\,x - 6\,x^{2} + 6\,x^{3} + 87\,n + 157\,n x + 44\,n x^{2} - 25\,n x^{3} - n x^{4} - 96\,n^{2} - 218\,n^{2} x - 99\,n^{2} x^{2} + 20\,n^{2} x^{3} + n^{2} x^{4} - 27\,n^{3} + 43\,n^{3} x + 69\,n^{3} x^{2} + 3\,n^{3} x^{3} + 54\,n^{4} + 48\,n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x - 9\,x^{2} + 3\,x^{4} + 23\,n x + 42\,n x^{2} + 10\,n x^{3} - 10\,n x^{4} - n x^{5} - 9\,n^{2} x - 37\,n^{2} x^{2} - 31\,n^{2} x^{3} - 3\,n^{2} x^{4} - 18\,n^{3} x - 20\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	2														7
2	35	32	6													73
3	252	378	165	20												815
4	1815	3840	2772	784	70											9281
5	13013	35750	36630	17160	3570	252										106375
6	92820	314496	420420	280280	96525	15840	924									1221305
7	658920	2658936	4399668	3836560	1881880	510510	69069	3432								14018975
8	4657983	21829632	43139880	46667040	29980860	11583936	2586584	297440	12870							160756225
9	32807775	175215150	403038108	521694096	416340540	210815640	67016040	12697776	1268982	48620						1840942727
10	230345115	1381380000	3627602550	5474211600	5234101950	3294910080	1372879200	370080480	60843510	5374720	184756					21051913961
11	1612806000	10733223930	31697490825	54681927300	61022461500	46062116100	23870937090	8433400800	1971062280	286016500	22632610	705432				240394780367
12	11265099300	82396154880	270370797840	525129633600	670604734800	591285895200	367630803540	161571730320	49478811525	10198531200	1323847800	94853472	2704156			2741353597633
13	78516779292	626139252648	2260529926200	4883211933600	7027157022600	7095636765360	5156205294240	2719674757800	1036102342275	279719089650	51534452970	6049583280	395965700	10400600		31220883566215
14	546222367160	4717023368960	18584109415872	44207654216544	70811162522100	80628070723200	67134985117200	41402923436160	18933407012520	6358326374400	1533697865700	255328000200	27349157745	1647455040	40116600	355141947149401

Technical details

Node id: n-649fb22a61937772 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-93693dbe17940c80

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:        5,         2;
  2:       35,        32,         6;
  3:      252,       378,       165,        20;
  4:     1815,      3840,      2772,       784,        70;
  5:    13013,     35750,     36630,     17160,      3570,       252;
  6:    92820,    314496,    420420,    280280,     96525,     15840,      924;
  7:   658920,   2658936,   4399668,   3836560,   1881880,    510510,    69069,     3432;
  8:  4657983,  21829632,  43139880,  46667040,  29980860,  11583936,  2586584,   297440,   12870;
  9: 32807775, 175215150, 403038108, 521694096, 416340540, 210815640, 67016040, 12697776, 1268982, 48620;

%F Formula

The row polynomials satisfy (8*n^4-8*n^3-2*n^2+2*n+(12*n^3-18*n^2+6*n)*x+(4*n^2-2*n)*x^2)*P(n,x) = (54*n^4-27*n^3-96*n^2+87*n-18+(48*n^4+43*n^3-218*n^2+157*n-30)*x+(2*n^4+69*n^3-99*n^2+44*n-6)*x^2+(3*n^3+20*n^2-25*n+6)*x^3+(n^2-n)*x^4)*P(n-1,x) - ((18*n^3+9*n^2-23*n+6)*x+(20*n^3+37*n^2-42*n+9)*x^2+(2*n^3+31*n^2-10*n)*x^3+(3*n^2+10*n-3)*x^4+n*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*P[n - 1, x] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*D[P[n - 1, x], x])/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*v[i - 1] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*deriv(v[i - 1], x))/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*x^2)); 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) = binomial(2*n, k) * binomial(3*n - k, 2*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

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.