A204420 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-3\,x - 2\,n + 6\,n x + 4\,n^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	24	54	27													105
3	720	1980	1620	405												4725
4	40320	126000	132300	56700	8505											363825
5	3628800	12428640	15309000	8675100	2296350	229635										42567525
6	479001600	1760330880	2430933120	1650310200	589396500	106091370	7577955									7023641625
7	87178291200	339061282560	511082732160	395162848080	171632260800	42295092840	5516751240	295540245								1552224799125
8	20922789888000	85297730918400	137917613433600	117837806486400	58974070755600	17874274017600	3227299475400	319183464600	13299311025							442384067750625
9	6402373705728000	27168167945318400	46552973987520000	43092167069952000	24055793782020000	8477205457921200	1899141614370000	262262413413000	20347945868250	678264862275						157931112186973125
10	2432902008176640000	10688839120447488000	19238715688140748800	19028543003870400000	11597455160154864000	4592518319585196000	1204874524562108400	207910789116030000	22681176994476000	1417573562154750	38661097149675					69015896025707255625
11	1124000727777607680000	5091516500161867776000	9561683512509217689600	10003225956140991974400	6556822493235382368000	2852383138738116984000	845980684481561428800	171961879619018689200	23577083485757802000	2083833136367482500	107168561298899100	2435649120429525				36233345413496309203125
12	620448401733239439360000	2888073158306005942272000	5629363937416257041203200	6181536890152963590451200	4309588607239659513369600	2026936244616681958560000	663795774406751980593600	153295624778926055025600	24879919777850596258800	2777094651792138678000	202941532246348595700	8739109044101135700	168059789309637225			22500907501781208015140625
13	403291461126605635584000000	1923781183028896820428800000	3875692046193517522452480000	4440201273905645611883520000	3264847861467250952974080000	1640727704543817736566720000	583487471710639934277840000	149426839186808334311160000	27669119714022341695140000	3671105507003684860110000	340194094844536988055000	20901035797141882882500	764672041358849373750	12604484198222791875		16313157938791375810976953125
14	304888344611713860501504000000	1487045182721101052726476800000	3085849462747639889428807680000	3670723218814343001902592000000	2827881286455599015010969600000	1504842821413973536035340800000	574015472681293026975951360000	160229175633788935415742000000	33021428477932365400729800000	5016554460130595431549500000	554546281769768436638490000	43356904745046759491625000	2271075962835782640037500	71467425403923229931250	1020963220056046141875	13654113194768381553787709765625

Technical details

Node id: n-81c130d5be6f4626 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-570085136c45352b

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,                 3;
  2:               24,                54,                27;
  3:              720,              1980,              1620,               405;
  4:            40320,            126000,            132300,             56700,              8505;
  5:          3628800,          12428640,          15309000,           8675100,           2296350,           229635;
  6:        479001600,        1760330880,        2430933120,        1650310200,         589396500,        106091370,          7577955;
  7:      87178291200,      339061282560,      511082732160,      395162848080,      171632260800,      42295092840,       5516751240,       295540245;
  8:   20922789888000,    85297730918400,   137917613433600,   117837806486400,    58974070755600,   17874274017600,    3227299475400,    319183464600,    13299311025;
  9: 6402373705728000, 27168167945318400, 46552973987520000, 43092167069952000, 24055793782020000, 8477205457921200, 1899141614370000, 262262413413000, 20347945868250, 678264862275;

%F Formula

The row polynomials satisfy P(n,x) = (4*n^2-2*n+(6*n-3)*x)*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[(4*n^2 - 2*n + (6*n - 3)*x)*P[n - 1, x]];
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] = (4*n^2 - 2*n + (6*n - 3)*x)*v[i - 1]); 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.