A253283 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(4 - n\right)\,P_{n-3}(x) + \left(-23 - 28\,x + 7\,n + 8\,n x\right)\,P_{n-2}(x) + \left(12 + 12\,x - 7\,n - 8\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{3}(x) = 5 + 16\,x + 12\,x^{2}\)

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	2														3
3	5	16	12													33
4	25	114	168	80												387
5	129	768	1680	1600	560											4737
6	681	5010	14520	20720	14560	4032										59523
7	3653	32016	115500	219520	231840	129024	29568									761121
8	19825	201698	870576	2066400	2913120	2439360	1123584	219648								9854211
9	108545	1257472	6318144	17982720	31711680	35481600	24600576	9664512	1647360							128772609
10	598417	7777314	44583120	147950880	313236000	438774336	406678272	240514560	82368000	12446720						1694927619
11	3317445	47800080	307844460	1166943360	2883360480	4852463616	5633147520	4454461440	2296419840	697016320	94595072					22437369633
12	18474633	292292946	2089284120	8906005680	25155530400	49436202816	68976940032	68332492800	47104611840	21520378880	5864894464	722362368				298419470979
13	103274625	1779856128	13981617936	66197651520	210391861680	472884572160	770760646656	917940621312	792824947200	484326768640	198649651200	49120641024	5538111488			3984500221569
14	579168825	10799942322	92479560264	481490689680	1700579200320	4302692634240	8023407135744	11164961753088	11594399731200	8874673167360	4866916454400	1810962456576	409820250112	42600857600		53376363001731
15	3256957317	65336473104	605695397580	3439276888320	13363518168000	37587441475584	78922761485568	125674282905600	152510786588160	140365843537920	96450365147136	47982197932032	16337428889600	3408068608000	328635187200	717044895641121

Technical details

Node id: n-57fab0d047111899 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c15c2b3e5c2d6a7d

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,       2;
   3:      5,      16,       12;
   4:     25,     114,      168,        80;
   5:    129,     768,     1680,      1600,       560;
   6:    681,    5010,    14520,     20720,     14560,      4032;
   7:   3653,   32016,   115500,    219520,    231840,    129024,     29568;
   8:  19825,  201698,   870576,   2066400,   2913120,   2439360,   1123584,    219648;
   9: 108545, 1257472,  6318144,  17982720,  31711680,  35481600,  24600576,   9664512,  1647360;
  10: 598417, 7777314, 44583120, 147950880, 313236000, 438774336, 406678272, 240514560, 82368000, 12446720;

%F Formula

The row polynomials satisfy -(n-1)*P(n,x) = -(n-4)*P(n-3,x) + (7*n-23+(8*n-28)*x)*P(n-2,x) - (7*n-12+(8*n-12)*x)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 5 + 16*x + 12*x^2.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 5 + 16*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 4)*P[n - 3, x] + (7*n - 23 + (8*n - 28)*x)*P[n - 2, x] - (7*n - 12 + (8*n - 12)*x)*P[n - 1, 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 + 2*x; v[3] = 5 + 16*x + 12*x^2; for(n = 4, N, my(i = n); v[i] = (-(n - 4)*v[i - 3] + (7*n - 23 + (8*n - 28)*x)*v[i - 2] - (7*n - 12 + (8*n - 12)*x)*v[i - 1])/(-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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) 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

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.