A204420 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 54\,x - 27\,x^{2} + 108\,n - 180\,n x + 72\,n x^{2} - 144\,n^{2} + 168\,n^{2} x - 36\,n^{2} x^{2} + 80\,n^{3} - 48\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 6\,x - 12\,n + 12\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	45	90	27													162
3	1260	3465	2430	405												7560
4	66465	217980	217350	79380	8505											589680
5	5681340	20947815	25685100	13522950	3061800	229635										69128640
6	716911965	2880184230	4065806745	2677544100	879884775	136403190	7577955									11364312960
7	125372307060	538456083165	843739426530	649163269755	269707838400	60914128275	6895939050	295540245								2494544532480
8	29030535158625	131614360277400	223777162709100	193451011954200	94844447047350	27280334881800	4509944138700	390113123400	13299311025							704911208601600
9	8599375008724500	40788114358466475	74104595111547000	70217301947492700	39018216160123200	13340063069383050	2823390533363400	357219494131500	24417535041900	678264862275						249273371483136000
10	3172059140515390125	15637153272035144850	30027902806351625175	30663321413972247000	18815830425171029250	7339035117327813900	1856485838632575150	301900211964351000	30220090938662625	1675314209819250	38661097149675					107845622292215808000
11	1425884301151259449500	7268657824488465630225	14633599368974139303750	15909484054217011283025	10588717420386963858000	4587161876123238061650	1330587657194577081300	259730946719830035450	33519944450711218500	2723867599680352125	125029988182048950	2435649120429525				56040194726943178752000
12	767336799657874920098625	4028848018364413451106900	8450809144806586769060850	9693042310383232135034700	6906822792750116715453975	3261827059352345730846600	1055714580871109891631900	237278363101751711539800	36884897360986065566175	3873964876686370966500	260866139628403559250	10083587358578233500	168059789309637225			34442709049311880052736000
13	487085802286765561066687500	2625630228925845908951559375	5709976138833239802411382500	6861213417819581406997661250	5183174140074859131087885000	2632369619177267831409673125	932612147800141767723765000	234766868826936788189347500	42162260446950500814457500	5349044787599728265675625	466498495834698679852500	26472217812756356891250	873910904410113570000	12604484198222791875		24714833526208224490291200000
14	360148254054952865821021828125	1987929632669496916715083518750	4464391055125258474574014783125	5588787870793520364364710817500	4442118001179416650354557920625	2401236825551212906674458291250	918232806787734980745724145625	253887341644325336840617725000	51229330000471913034374004375	7529954228200956079279001250	795141545854854765321129375	58538703454185354958057500	2838844953544728300046875	80996415457779660588750	1020963220056046141875	20476347673146232627888128000000

Technical details

Node id: n-a58c3ee71fbac741 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f681b7570d759557

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:                4,                 3;
  2:               45,                90,                27;
  3:             1260,              3465,              2430,               405;
  4:            66465,            217980,            217350,             79380,              8505;
  5:          5681340,          20947815,          25685100,          13522950,           3061800,            229635;
  6:        716911965,        2880184230,        4065806745,        2677544100,         879884775,         136403190,          7577955;
  7:     125372307060,      538456083165,      843739426530,      649163269755,      269707838400,       60914128275,       6895939050,       295540245;
  8:   29030535158625,   131614360277400,   223777162709100,   193451011954200,    94844447047350,    27280334881800,    4509944138700,    390113123400,    13299311025;
  9: 8599375008724500, 40788114358466475, 74104595111547000, 70217301947492700, 39018216160123200, 13340063069383050, 2823390533363400, 357219494131500, 24417535041900, 678264862275;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*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; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 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)

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.