A253283 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 14\,x + 7\,n + 4\,n x\right)\,P_{n-2}(x) + \left(-14\,x + 4\,n x\right)\,P_{n-2}'(x) + \left(12 + 6\,x - 7\,n - 4\,n x\right)\,P_{n-1}(x) + \left(6\,x - 4\,n x\right)\,P_{n-1}'(x)\)

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

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

\(P_{3}(x) = 5 + 11\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	5	11	3													19
4	25	109	72	10												216
5	129	1039	1265	410	35											2878
6	681	9761	19660	11200	2170	126										43598
7	3653	91307	287427	256760	84840	10962	462									735411
8	19825	854365	4062828	5353026	2686110	579810	53592	1716								13611272
9	108545	8013215	56318817	105397194	75427275	24201870	3681678	255684	6435							273410713
10	598417	75396673	771767928	2000574400	1963569300	878906028	195723528	22119240	1196910	24310						5909876734
11	3317445	711821131	10505204963	37060070960	48584890640	29154072948	8915822916	1458102360	127262850	5518370	92378					136526176961
12	18474633	6742547021	142457951152	675238372666	1160241588220	908401722228	366497963832	81351256272	10182997110	707153590	25126816	352716				3351865506256
13	103274625	64063973167	1928134500305	12162805590650	27010434694243	27058931559660	14012339961620	4054595976288	682601549916	67495523810	3818444630	113221836	1352078			87045439622828
14	579168825	610409873633	26077482042820	217345638328000	617177825646382	779614587842090	507984711914680	186386453593912	40528727702034	5350746691290	428480572520	20129737264	505677172	5200300		2381526283990922
15	3256957317	5830768239979	352690051101923	3862445572078184	13908079326148280	21903773124855622	17690836823678018	8066961752663528	2203491459873990	372983139839310	39642457788934	2623204385984	103974798200	2241329300	20058300	68409467173796869

Technical details

Node id: n-d58a953b91eff3a7 · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-be3c7f562b31849c

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      1,        1;
   3:      5,       11,         3;
   4:     25,      109,        72,         10;
   5:    129,     1039,      1265,        410,         35;
   6:    681,     9761,     19660,      11200,       2170,       126;
   7:   3653,    91307,    287427,     256760,      84840,     10962,       462;
   8:  19825,   854365,   4062828,    5353026,    2686110,    579810,     53592,     1716;
   9: 108545,  8013215,  56318817,  105397194,   75427275,  24201870,   3681678,   255684,    6435;
  10: 598417, 75396673, 771767928, 2000574400, 1963569300, 878906028, 195723528, 22119240, 1196910, 24310;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(4*n-14)*x)*P(n-2,x) + (4*n-14)*x*d/dx P(n-2,x) - (7*n-12+(4*n-6)*x)*P(n-1,x) - (4*n-6)*x*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 5 + 11*x + 3*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 5 + 11*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (4*n - 14)*x)*P[n - 2, x] + (4*n - 14)*x*D[P[n - 2, x], x] - (7*n - 12 + (4*n - 6)*x)*P[n - 1, x] - (4*n - 6)*x*D[P[n - 1, x], x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 5 + 11*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (4*n - 14)*x)*v[i - 2] + (4*n - 14)*x*deriv(v[i - 2], x) - (7*n - 12 + (4*n - 6)*x)*v[i - 1] - (4*n - 6)*x*deriv(v[i - 1], x))/(-n + 1)); 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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

References

[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.