A127952 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} - 4\,n - 12\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 12\,x + 4\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

\(P_{2}(x) = 6 + 27\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	6	27	27													60
3	16	96	180	108												400
4	40	300	810	945	405											2500
5	96	864	3024	5184	4374	1458										15000
6	224	2352	10080	22680	28350	18711	5103									87500
7	512	6144	31104	86400	142560	139968	75816	17496								500000
8	1152	15552	90720	299376	612360	796068	642978	295245	59049							2812500
9	2560	38400	253440	967680	2358720	3810240	4082400	2799360	1115370	196830						15625000
10	5632	92928	684288	2965248	8382528	16166304	21555072	19630512	11691702	4113747	649539					85937500
11	12288	221184	1797120	8709120	27993600	62705664	99937152	113374080	89754480	47239200	14880348	2125764				468750000
12	26624	519168	4612608	24710400	88957440	226841472	420323904	570439584	562933800	394053660	185768154	52966953	6908733			2539062500
13	57344	1204224	11612160	68124672	271434240	775982592	1638185472	2586608640	3055431456	2667440160	1673212464	714256704	186004350	22320522		13671875000
14	122880	2764800	28753920	183306240	800616960	2535287040	6004627200	10808328960	14861452320	15536972880	12159370080	6908733000	2694405870	645700815	71744535	73242187500

Technical details

Node id: n-bb688d94764393f4 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-371721bfd00b48c7

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    2,     6;
  2:    6,    27,     27;
  3:   16,    96,    180,    108;
  4:   40,   300,    810,    945,     405;
  5:   96,   864,   3024,   5184,    4374,    1458;
  6:  224,  2352,  10080,  22680,   28350,   18711,    5103;
  7:  512,  6144,  31104,  86400,  142560,  139968,   75816,   17496;
  8: 1152, 15552,  90720, 299376,  612360,  796068,  642978,  295245,   59049;
  9: 2560, 38400, 253440, 967680, 2358720, 3810240, 4082400, 2799360, 1115370, 196830;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(12*n-24)*x+(9*n-18)*x^2)*P(n-2,x) + (4*n-8+(6*n-12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 6*x, P(2,x) = 6 + 27*x + 27*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[2, x_] := 6 + 27*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*P[n - 2, x] + (4*n - 8 + (6*n - 12)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; v[3] = 6 + 27*x + 27*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (12*n - 24)*x + (9*n - 18)*x^2)*v[i - 2] + (4*n - 8 + (6*n - 12)*x)*v[i - 1])/(n - 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).?
  - 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.