A272866 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 20\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 20 + 32\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	P_n(1)
0																													0
1	20	32																											52
2	200	1120	1920	1024																									4264
3	1500	14400	52800	92160	76800	24576																							262236
4	10000	136000	768000	2329600	4096000	4177920	2293760	524288																					14335568
5	62500	1100000	8400000	36480000	99200000	175104000	200704000	144179200	58982400	10485760																			734697860
6	375000	8100000	78000000	441600000	1632000000	4131840000	7311360000	9043968000	7667712000	4246732800	1384120320	201326592																	36147134712
7	2187500	56000000	651000000	4547200000	21280000000	70425600000	169523200000	300482560000	392232960000	372506624000	250295091200	112742891520	30534533120	3758096384															1729037943724
8	12500000	370000000	5040000000	41888000000	237440000000	971980800000	2967552000000	6879641600000	12209356800000	16588472320000	17116954624000	13176825446400	7328287948800	2780991324160	644245094400	68719476736													81017777934496
9	70312500	2362500000	36900000000	355680000000	2368800000000	11563776000000	42835968000000	122963558400000	276885504000000	491854233600000	689291919360000	757843623936000	645805375488000	417712413081600	198105366528000	64939905515520	13142599925760	1236950581248											3736945007228628
10	390625000	14687500000	258750000000	2838000000000	21720000000000	123206400000000	537062400000000	1840619520000000	5032304640000000	11071553536000000	19681902592000000	28271915827200000	32682855628800000	30144630620160000	21864067891200000	12189117186048000	5037996638208000	1453416932966400	261134011596800	21990232555520									170238605884859720
11	2148437500	89375000000	1753125000000	21560000000000	186450000000000	1205424000000000	6046656000000000	24107212800000000	77623603200000000	203999150080000000	440393203712000000	783356736307200000	1148139339776000000	1382404010803200000	1358652506112000000	1078595137044480000	680677154488320000	333358181646336000	122080150421504000	31446032554393600	5079743720325120	387028092977152							7677761125407173372
12	11718750000	534375000000	11550000000000	157410000000000	1518000000000000	11019360000000000	62536320000000000	284461056000000000	1054918656000000000	3227108966400000000	8207920005120000000	17442146353152000000	31040861110272000000	46263834142310400000	57612624199680000000	59655807251251200000	50960431212134400000	35495843117137920000	19820071480197120000	8653706266411008000	2844656483382067200	661818038990929920	97108866965176320	6755399441055744					343405315790939027184
13	63476562500	3148437500000	74343750000000	1112150000000000	11833250000000000	95306640000000000	603803200000000000	3086786560000000000	12961428480000000000	45267402752000000000	132665517670400000000	328253114941440000000	688321782087680000000	1225478295781376000000	1852493046743040000000	2373315605404057600000	2567196978970624000000	2330445818835763200000	1759749462399385600000	1091570405044387840000	546453430713974784000	215216646802269798400	64172776252820684800	13612129873727324160	1829587348619264000	117093590311632896			15252919443047541790756
14	341796875000	18320312500000	469218750000000	7644000000000000	88952500000000000	787222800000000000	5507902400000000000	31268357120000000000	146680934400000000000	576104488960000000000	1912678383616000000000	5404920093081600000000	13062344068300800000000	27080175187394560000000	48230193482956800000000	73795175670874112000000	96860815423963136000000	108736081310829772800000	103904570379337728000000	83935357018780467200000	56775025626143784960000	31737222815847284736000	14396794147866411008000	5163827332743010713600	1408782258436833280000	274584469280779141120	34047213182920949760	2017612633061982208	673475058485329922145688

Technical details

Node id: n-90139221eebb3966 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-dc1f92b0a0065af1

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:       20,         32;
  2:      200,       1120,        1920,         1024;
  3:     1500,      14400,       52800,        92160,         76800,          24576;
  4:    10000,     136000,      768000,      2329600,       4096000,        4177920,        2293760,          524288;
  5:    62500,    1100000,     8400000,     36480000,      99200000,      175104000,      200704000,       144179200,        58982400,        10485760;
  6:   375000,    8100000,    78000000,    441600000,    1632000000,     4131840000,     7311360000,      9043968000,      7667712000,      4246732800,      1384120320,       201326592;
  7:  2187500,   56000000,   651000000,   4547200000,   21280000000,    70425600000,   169523200000,    300482560000,    392232960000,    372506624000,    250295091200,    112742891520,     30534533120,      3758096384;
  8: 12500000,  370000000,  5040000000,  41888000000,  237440000000,   971980800000,  2967552000000,   6879641600000,  12209356800000,  16588472320000,  17116954624000,  13176825446400,   7328287948800,   2780991324160,    644245094400,    68719476736;
  9: 70312500, 2362500000, 36900000000, 355680000000, 2368800000000, 11563776000000, 42835968000000, 122963558400000, 276885504000000, 491854233600000, 689291919360000, 757843623936000, 645805375488000, 417712413081600, 198105366528000, 64939905515520, 13142599925760, 1236950581248;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+20*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 20 + 32*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 20 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 20*n*x + 16*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 20 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 20*n*x + 16*n*x^2)*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 - 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → D → basis A013611

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.